{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Headi ng 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Outp ut" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Heading 2" -1 256 1 {CSTYLE "" -1 -1 "Ti mes" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 53 "REPRESENTATION THEORY\nAND\nINVA RIANTS\nOF\nFINITE GROUPS" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 26 "NOTE S, WARNINGS and ERRATA" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "# \+ Applied Algebra package created at the University of California, San D iego by Adriano Garsia." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " # Modifications and examples by Jason Bandlow and Gregg Musiker." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 83 "# All the Qual Questions from 2002 and procedures c omplete. Execute Worksheet Once" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "# Takes about 57 Maple secs to execute--much of this time is spent in the\n# 'Old Qual Questions' section." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "# For a couple procedures it is still unclear what t hey do, but they are procedures we haven't seen from class: see UNCLEA R PROCS" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "WARNINGS" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "# Sometimes, variables are written as x||i (|| is co ncatentation), sometimes as x[i]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# Maple treats these differently, so you should know which is being used." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "# RULE OF THUMB: Do not \+ assign a variable a 1 letter name. If you do, you may have to use unas sign." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "# For example, t o use the procedure testAI, the variables y1,...,y_n must be unassigne d (use unassign('y1','y2','y3','y4');)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "# Also names corresponding to certain greek letters (sigma, etc.) are protected my Maple. In general,\n# you should use abbreviations: (sig, la, etc.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "# This package will fail miserably if the file SF.mpl is not in the directory D:\\My docu~1\\ \n# If this file is somewhere else, make the obvious change \+ in the 'Call packages' section below" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "# In \+ order to re-execute the worksheet more than once in the same session, \+ a line in the 'Call packages' section must" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "# be commented out. See that section for more det ails." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "UNCLEAR PROCEDURES " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 65 "# WHAT ARE DESCMON? (descent monomials) (Perm utation procedures)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "# \+ Do IDEMPREY and IDEMPHILB sum over partitions or perms (i.e. should we use n! or h_lambda (invariant theory procedures)" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 84 "# MKBASIS doesn't work, USES INERT PROCEDUR E Testmon (invariant theory procedures)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# What is mkgram (the Gramm matrix?) (Gordan Basis \+ Procedures)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "# How does mkdual(alpha) (which makes the dual to the artin basis) depend on Alp ha?\n# (Gordan Bases Procedures)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "MAC CONVERS ION ISSUES" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "# LIST TOOLS \+ is not a package on the MACS" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "# ON the MACS, you should have the SF package as folder in sam e dierctory as this worksheet and\n# delete the next subsection " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "# Big conversions issues fi xed but there are a couple randomn errors that will need to be fixed \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 19 "MY RECENT ADDITIONS" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "# Changed Allinvar and added those other new procedu res SEE NEW PROCEDURES SECTION since (5/20/03) got rid of Allinvar and replaced with Allivar2 and Allinvar3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "# Changed and Completed Old Qual Questions in Part II I (but left R.2, R.4, S1. - S.5 alone" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "NEW PROCEDU RES" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1018 "# I created several new procedures in the Invariant Theory Procs section:\n\n# getmons (n,d) outputs a list of monomials of degree d assuming n variables (Ga rsia also wrote a code for this proc)\n# Allinvar(G,d,m) creates mi nimal set of quasi-generators homogeneous of degree d assuming group G given by m x m matrices\n# Allivar2 is a procedure that does rudi mentary Jacobian test to create set of quasi-generators w/o repeats an d obvious dependencies\n# only works for 2 x 2 matrices but wor ks faster once the degree gets large enough\n# Allinvar3 is the ana logous procedure for 3 x 3 matrices\n# Everyinvar creates the spann ing set of all invariants VERY QUICKLY by just applying the Reynolds o perator to all monomials.\n# but creates a set that is FAR FROM MINIMAL\n# Sep Poly(eta,ideal,n) returns the polynomial satisfied \+ by eta and the elements of the ideal assuming n variables\n# wo rks by creating Grobener basis and then selecting the GB element that \+ consists of only y_i's and y\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "# findbas(GB, n, d) outputs Quotient for basis Q[x1,...,x_ n] / GB (stops at degree d) (Another Proc from Prof. Garsia)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Call pack ages" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# Change the followi ng line based on where 'SF' is." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "read(\"D:\\\\MYDOCU~1\\\\SF.mpl\");" }}{PARA 6 "" 1 "" {TEXT -1 55 "SF 2.3v loaded. Run 'withSF()' to use abbreviated names" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "with(SF);" }}{PARA 7 "" 1 "" {TEXT -1 73 "Warning, the protected name conjugate has been redefined \+ and unprotected\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7:%$ParG%*add_bas isG%(char2sfG%*conjugateG%)dominateG%+dual_basisG%'evalsfG%&hooksG%(it ensorG%*jt_matrixG%&omegaG%)plethysmG%'scalarG%(sf2charG%%skewG%&stdeg G%'subParG%&thetaG%$toeG%$tohG%$topG%$tosG%'varsetG%$zeeG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "dual_basis(m,h); # Only run this \+ ONCE while Maple is open. If run again, re-run SF but don't run this \+ line" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%OkayG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " with(numtheory);" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, the protecte d name order has been redefined and unprotected\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7Q%&GIgcdG%)bigomegaG%&cfracG%)cfracpolG%+cyclotomicG%) divisorsG%)factorEQG%*factorsetG%'fermatG%)imagunitG%&indexG%/integral _basisG%)invcfracG%'invphiG%*issqrfreeG%'jacobiG%*kroneckerG%'lambdaG% )legendreG%)mcombineG%)mersenneG%(migcdexG%*minkowskiG%(mipolysG%%mlog G%'mobiusG%&mrootG%&msqrtG%)nearestpG%*nthconverG%)nthdenomG%)nthnumer G%'nthpowG%&orderG%)pdexpandG%$phiG%#piG%*pprimrootG%)primrootG%(quadr esG%+rootsunityG%*safeprimeG%&sigmaG%*sq2factorG%(sum2sqrG%$tauG%%thue G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdecompG%)QRde compG%*WronskianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(augmentG% (backsubG%%bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)charpolyG%) choleskyG%$colG%'coldimG%)colspaceG%(colspanG%*companionG%'concatG%%co ndG%)copyintoG%*crossprodG%%curlG%)definiteG%(delcolsG%(delrowsG%$detG %%diagG%(divergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenvectorsG%+ eigenvectsG%,entermatrixG%&equalG%,exponentialG%'extendG%,ffgausselimG %*fibonacciG%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG%(geneqns G%*genmatrixG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG%+htransp oseG%)ihermiteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG%'ismithG% *issimilarG%'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG%*leastsq rsG%)linsolveG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'mulrowG%) multiplyG%%normG%*normalizeG%*nullspaceG%'orthogG%*permanentG%&pivotG% *potentialG%+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG%'rowdimG% )rowspaceG%(rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smithG%,stackm atrixG%*submatrixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG%*sylvest erG%)toeplitzG%&traceG%*transposeG%,vandermondeG%*vecpotentG%(vectdimG %'vectorG%*wronskianG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "wi th(combinat);" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, the assigned na me fibonacci now has a global binding\n" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, the protected name Chi has been redefined and unprotected \n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7C%$ChiG%%bellG%)binomialG%)cart prodG%*characterG%'chooseG%,compositionG%)conjpartG%+decodepartG%+enco departG%*fibonacciG%*firstpartG%)graycodeG%)inttovecG%)lastpartG%,mult inomialG%)nextpartG%)numbcombG%)numbcompG%)numbpartG%)numbpermG%*parti tionG%(permuteG%)powersetG%)prevpartG%)randcombG%)randpartG%)randpermG %-setpartitionG%*stirling1G%*stirling2G%(subsetsG%)vectointG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(grobner);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7*%(finduniG%'finiteG%'gbasisG%'gsolveG%(leadmon G%(normalfG%)solvableG%&spolyG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(ListTools);" }}{PARA 7 "" 1 "" {TEXT -1 58 "Warning, the \+ assigned name Group now has a global binding\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#77%,BinaryPlaceG%-BinarySearchG%+CategorizeG%+DotProduc tG%*EnumerateG%0FindRepetitionsG%(FlattenG%,FlattenOnceG%&GroupG%+Inte rleaveG%%JoinG%-JoinSequenceG%+MakeUniqueG%,OccurrencesG%$PadG%,Partia lSumsG%(ReverseG%'RotateG%'SortedG%&SplitG%*TransposeG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "q-binomial coefficients" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "qan(n) : Returns (1-q)*(1-q^2)*...*(1-q^n)" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 94 "qan:=proc(k)\nlocal out,i;\nout:=1;\nfor i from 1 t o k do\n out:=out*(1-q^i);\n od;\nout;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "qan(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* ,,&\"\"\"F%%\"qG!\"\"F%,&F%F%*$)F&\"\"#F%F'F%,&F%F%*$)F&\"\"$F%F'F%,&F %F%*$)F&\"\"%F%F'F%,&F%F%*$)F&\"\"&F%F'F%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "qbin(n,k) : Returns (qan(n) / [ (qan(k) * qan(n-k) ]" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "qbin:=(n,k)->qan(n)/qan(k)/qan(n-k);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%qbinGf*6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF) *(-%$qanG6#9$\"\"\"-F/6#9%!\"\"-F/6#,&F1F2F5F6F6F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "qbin(6,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.,&\"\"\"F%%\"qG!\"\"F',&F%F%*$)F&\"\"#F%F'F',&F%F%*$) F&\"\"$F%F'F',&F%F%*$)F&\"\"%F%F'F%,&F%F%*$)F&\"\"&F%F'F%,&F%F%*$)F&\" \"'F%F'F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "qfac(n) : Returns (1-q^1)*(1-q^2)*...*(1- q^n) (Synonym for qan(n))" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "qfac:=n->convert([seq(1-q^i,i=1..n)],`*`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%qfacGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%(convertG6 $7#-%$seqG6$,&\"\"\"F4)%\"qG%\"iG!\"\"/F7;F49$%\"*GF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "qfac(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.,&\"\"\"F%%\"qG!\"\"F%,&F%F%*$)F&\"\"#F%F'F%,&F%F%*$) F&\"\"$F%F'F%,&F%F%*$)F&\"\"%F%F'F%,&F%F%*$)F&\"\"&F%F'F%,&F%F%*$)F&\" \"'F%F'F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Auxiliary list and vector procedures" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 95 "conca(L1,L2) : Returns a list of every concatenation of an element of L1 with an element of L2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "conca:=proc(A,B)\nlocal a,b,out;\nout:=NULL;\nfor a \+ in A do\n for b in B do\n out:=out,[op(a),op(b)];\n od;od;\n[out]; \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "conca([1,2],[x1,x 2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$\"\"\"%#x1G7$F%%#x2G7$\"\" #F&7$F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "conca([ [1,2], [2,1] ], [ [3,4],[4,3] ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7&\" \"\"\"\"#\"\"$\"\"%7&F%F&F(F'7&F&F%F'F(7&F&F%F(F'" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 80 "ob (list) : Returns every element of the list, comma separated but not in a list." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "ob := proc (seq) `$`('seq[i]',('i') = 1 .. nops(seq)) end:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "ob([5,2,3,1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 '\"\"&\"\"#\"\"$\"\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 116 "equa(A,B) : Returns true i f lists L1 and L2 are (literally) equal, false o/w. (L1,L2 can be par titions or tableaux)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "equa :=proc(A,B)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "local out,h,k,i;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "h:=nops(A); k:=nops(B); out:=true; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "if h<>k then out:=false" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "else for i from 1 to k do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " if A[i]<>B[i] then out:=false" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "out;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "equa([3,2,1] ,[3,2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "equa([[1,2,3],[4,5],[6]],[[1,2,3],[ 4,5],[6]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 135 "lexo(pair1,pair2) : Returns true if pair1 < pair2 in lex order , false o/w. (pair1, pair2 are ordered pairs: [a,b]) (For use with sor t) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "lexo := proc (a, b) \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "local out; out := false;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "if a[1] < b[1] or a[1] = b[1] and a [2] < b[2] then out := true fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "o ut" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "oplist:=[[3,2],[5,8],[7,4],[1,8]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sort(oplist,lexo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$\"\"\"\"\")7$\"\"$\"\"#7$\"\"&F&7$\"\"(\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "par(list) : Returns the list in reverse sorted order " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "par := proc (sec) " } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "local n, out, i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "n := nops(sec); out := sort(sec);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "[seq(out[n+1-i],i = 1 .. n)]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "par([1,3,2,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"$F$\"\"# \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 103 "parmindelt(list) : Returns the list in r everse sorted oreder with delta = [n,n-1,...,2,1,0] subtracted " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "parmindelt:=proc(q)\nlocal \+ sq,out,i,n,d,p;\nn:=nops(q);\nd:=[seq(i-1,i=1..n)];\nsq:=sort(q);\np:= sq-d;\nout:=NULL;\nfor i from 1 to n do\n if p[n+1-i]>0 then out:=out ,p[n+1-i]; fi;\n od;\n[out];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "par([2,6,8,5,4,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7(\"\")\"\"'\"\"&\"\"%\"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "parmindelt([2,6,8,5,4,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"$\"\"#F%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "abv(a,b) : \+ Returns 1 if a >= b, 0 otherwise" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "abv:=(a,b)->if a>=b then 1 else 0 fi: " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "abv(7,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 137 "locinG (sig, G) : Locates element sig of a gro up G if the group is entered as a subgroup of a permutation group with a canonical ordering" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "lo cinG:=proc(sig,G)\nlocal i,n,out,x;\noptions remember; \nfor i fro m 1 to nops(G) do\n x[op(G[i])]:=i;\n od;\nout:=x[op(sig) ];\nend: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "rever(v) : Returns vector v in re verse order" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "rever:=proc( v)\nlocal i,out,n;\nout:=NULL;\nn:=nops(v);\nfor i from 1 to n do\n o ut:=out,v[n+1-i];\n od;\n[out];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rever([2,3,4,6,9]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7'\"\"*\"\"'\"\"%\"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 107 "SCP(A,B) Returns the \+ usual scalar product of A and B if they are vectors of equal length (w ritten as lists)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "SCP:=pr oc(A,B)\nlocal out,i;\nout:=0;\nfor i from 1 to nops(B) do\n out:=ou t+A[i]*B[i];\n od;\nout;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "SCP([1,2,3,4],[x,y,z,w]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"xG\"\"\"*&\"\"#F%%\"yGF%F%*&\"\"$F%%\"zGF%F%*&\"\" %F%%\"wGF%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 27 "Auxiliary matrix procedures" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "mkzmat(n) : Returns the nxn zero matrix" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "mkzmat:=proc(n)\nlocal i,j,rw,out;\nout: =NULL;\nfor i from 1 to n do\n rw:=[seq(0,j=1..n)];\n out:=out,rw; \n od;\nmatrix([out]);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "mkzmat(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7' \"\"!F(F(F(F(F'F'F'F'Q(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "dotprod(A,B ) : Returns the dot product of the matrices A and B" }{TEXT 256 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "dotprod:=proc(A,B)\nlocal \+ i,j,dim,out;\ndim:=rowdim(A);\nout:=0;\nfor i from 1 to dim do\n for j from 1 to dim do\n out:=out+A[i,j]*B[i,j];\n od;\n od;\nout; \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "AA:=matrix(2,2,[1 ,2,3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AAGK%'matrixG6#7$7$\" \"\"\"\"#7$\"\"$\"\"%Q(pprint16\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "BB:=matrix(2,2,[a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BBGK%'matrixG6#7$7$%\"aG%\"bG7$%\"cG%\"dGQ(pprint26 \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dotprod(AA,BB);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"aG\"\"\"*&\"\"#F%%\"bGF%F%*&\"\"$ F%%\"cGF%F%*&\"\"%F%%\"dGF%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 63 "mu(A,B) : Returns A * \+ B, for A,B matrices of the same dimension" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "mu:=proc(A,B)\nmultiply(A,B);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "AA:=matrix(2,2,[1,2,3,4]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#AAGK%'matrixG6#7$7$\"\"\"\"\"#7$\"\"$\"\"%Q(p print36\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "BB:=matrix(2,2 ,[a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BBGK%'matrixG6#7$7$ %\"aG%\"bG7$%\"cG%\"dGQ(pprint46\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "mu(AA,BB);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7$,&%\"aG\"\"\"*&\"\"#F*%\"cGF *F*,&%\"bGF**&F,F*%\"dGF*F*7$,&*&\"\"$F*F)F*F**&\"\"%F*F-F*F*,&*&F5F*F /F*F**&F7F*F1F*F*Q(pprint56\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "pow(A,n) : Returns the matrix A^n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "pow:=proc(A, n)\nlocal i,out;\nout:=multiply(A,inverse(A));\nfor i from 1 to n do\n out:=multiply(out,A);\n od;\nmatrix(out);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=matrix(2,2,[1,1,0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGK%'matrixG6#7$7$\"\"\"F*7$\"\"!F*Q(ppr int66\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "B:=pow(A,3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BGK%'matrixG6#7$7$\"\"\"\"\"$7$\" \"!F*Q(pprint76\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "mkmatA(n) : Returns the matrix || a_(i,j)|| i,j=1..n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "mkmat A:=proc(n)\nlocal i,j,out,rw;\nout:=NULL;\nfor i from 1 to n do\n rw: =NULL;\n for j from 1 to n do\n rw:=rw,cat(cat(a,i),j);\n od;\n \+ out:=out,[rw];\n od;\narray([out]);\nend:\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "AAA:=mkmatA(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AAAGK%'matrixG6#7%7%%$a11G%$a12G%$a13G7%%$a21G%$a22G%$a23G7%% $a31G%$a32G%$a33GQ(pprint86\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 67 "mkpermat(sig) : Return s the permutation matrix corresponding to sig" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "mkpermat:=proc(sig)\nlocal i,j,out,row,n;\nn:=n ops(sig);\nout:=NULL;\nfor i from 1 to n do\n row:=NULL;\n for j fro m 1 to n do\n if i=sig[j] then row:=row,1; else row:=row,0; fi;\n \+ od;\n out:=out,[row];\n od;\narray([out]);\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Z:=mkpermat([2,3,4,5,1]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZGK%'matrixG6#7'7'\"\"!F*F*F*\"\" \"7'F+F*F*F*F*7'F*F+F*F*F*7'F*F*F+F*F*7'F*F*F*F+F*Q(pprint96\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 79 "mycharpol(A) : Returns the determinant of (I-qA) (cha rpoly returns det(qI-A))" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "mycharpol:=proc(A)\nlocal d,out,cp;\ncp:=charpoly(A,q);\nd:=degree(cp );\nout:=expand(q^d*subs(q=1/q,cp));\nend:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "B:=matrix(2,2,[1,2,3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BGK%'matrixG6#7$7$\"\"\"\"\"#7$\"\"$\"\"%Q)pprint10 6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "mcpB:=mycharpol(B); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mcpBG,(\"\"\"F&*&\"\"&F&%\"qGF& !\"\"*&\"\"#F&)F)F,F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " cpB:=charpoly(B,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cpBG,(*$)%\" qG\"\"#\"\"\"F**&\"\"&F*F(F*!\"\"F)F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Id:=matrix([[1,0],[0,1]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "det(Id-q*B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,( \"\"\"F$*&\"\"&F$%\"qGF$!\"\"*&\"\"#F$)F'F*F$F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "mkGm at(a,b,c,d): Returns the matrix [a b], [c d]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "mkGmat:=(a,b,c,d)->matrix([[a,b],[c,d]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'mkGmatGf*6&%\"aG%\"bG%\"cG%\"dG6\"6 $%)operatorG%&arrowGF+-%'matrixG6#7$7$9$9%7$9&9'F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "mkGmat(1,2,3,4);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7$7$\"\"\"\"\"#7$\"\"$\"\"%Q)pprint116\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 72 "mkdihedr(n) : Returns, as a list of 2x2 matrices, \+ the dihedral group D_n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "m kdihedr:=proc(n)\nlocal i,out1,out2,A,R;\nA:=mkGmat(convert(cos(2*Pi/n ),radical),convert(-sin(2*Pi/n),radical),convert(sin(2*Pi/n),radical), convert(cos(2*Pi/n),radical));\nout1:=map(matrix,[seq(pow(A,i-1),i=1.. n)]);\nR:=mkGmat(1,0,0,-1);\nout2:=seq(multiply(R,pow(A,i-1)),i=1..n); \n[op(out1),out2];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "D3:=mkdihedr(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#D3G7(K%'matrix G6#7$7$\"\"\"\"\"!7$F,F+Q)pprint126\"KF'6#7$7$#!\"\"\"\"#,$*&F6F5\"\"$ #F+F6F57$,$*&F6F5F9F:F+F4Q)pprint13F/KF'6#7$7$F4F<7$F7F4Q)pprint14F/KF '6#7$F*7$F,F5Q)pprint15F/KF'6#7$F37$F7F:Q)pprint16F/KF'6#7$FB7$F " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "Vand(n) : Returns the Vandermonde determi nant of order n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "Vand := \+ proc(n) local i,j,V; V := 1; for i from 1 to n-1 do; for j from i+1 to n do; V := V*(cat(x,i)-cat(x,j)); od; od; V; end; " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%VandGf*6#%\"nG6%%\"iG%\"jG%\"VG6\"F,C%>8&\"\"\"?(8 $F0F0,&9$F0F0!\"\"%%trueG?(8%,&F2F0F0F0F0F4F6>F/*&F/F0,&-%$catG6$%\"xG F2F0-F>6$F@F8F5F0F/F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 90 "DELTLA(lamda) : returns th e matrix defined as Delta[lambda] = ||x_i^(p_j+n-j)|| i,j=1..n " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "DELTLA := proc(la) local f, n, M,i,j; n := nops(la); f := (i,j) -> (cat(x,i))^(la[j]+n-j); M := m atrix(n,n,f) end; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DELTLAGf*6#%# laG6'%\"fG%\"nG%\"MG%\"iG%\"jG6\"F.C%>8%-%%nopsG6#9$>8$f*6$F,F-F.6$%)o peratorG%&arrowGF.)-%$catG6$%\"xGF5,(&T$6#9%\"\"\"T%FGFF!\"\"F.F.6&F'F 5F*F1>8&-%'matrixG6%F1F1F7F.F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Partition procedures" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "size(la) : Returns |lambda|" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "size:=lambda->convert(lambda,`+`): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "size([3,2,2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 81 "count(la) \+ : Returns the # of parts of size i in lambda, for i from 1 to |lambda| " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "count:=proc(la)\nlocal \+ i,k,out,n;\nn:=size(la);\nout:=[seq(0,i=1..n)];\nfor i from 1 to nops( la) do\n out[la[i]]:=out[la[i]]+1;\n od;\nout;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "count([3,3,1]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7)\"\"\"\"\"!\"\"#F%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "size([3,3,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"(" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "conj(la) : Returns the conjugate partition to lambda" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "conj:=proc(lambda)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "local \+ out,j;\nout:=NULL;\nfor j from 1 to lambda[1] do\n out:=out,convert( map(abv,lambda,j),`+`);\n od;\n[out];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "conj([3,2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"$F$\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "hooks(la) : Returns \+ a list of the hooks of lambda" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "hooks:=proc(la)\nlocal cla,i,j,out;\nout:=NULL;\ncla:=conj(la); \nfor i from 1 to nops(la) do\n for j from 1 to la[i] do\n out: =out,1+cla[j]-i+la[i]-j;\n od;od;\n[out];\nend:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 15 "hooks([3,2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"&\"\"$\"\"\"F%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "hla(la) : R eturns the product of the hooks of lambda" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "hla:=lambda->convert(hooks(lambda),`*`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "hla([3,2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#X" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 88 "fla(la) : Returns the number of \+ standard tableaux of shape lambda (which is n!/hla(la))." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "fla:=proc(lambda)\nlocal n;\nn:=con vert(lambda,`+`);\nn!/hla(lambda);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "fla([3,2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "6!/hla([3,2,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 118 "corns(la) \+ : Returns a list of the places a cell could legally be added to a part ition lambda (the 'corners' of lambda)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "corns:=proc(la)\nlocal i,te,out,k;\nk:=nops(la);\nou t:=[1,la[1]+1];\nfor i from 1 to k-1 do\n if la[i]>la[i+1] then out:= out,[i+1,la[i+1]+1];fi;\n od;\nout:=out,[k+1,1];\n[out];\nen d:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "corns([3,2,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$\"\"\"\"\"%7$\"\"#\"\"$7$F)F(7$F&F %" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 71 "mke(la) : Returns the elementary function correspo nding to partition la" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "mk e:=proc(la)\nlocal i,out;\nout:=1;\nfor i from 1 to nops(la) do\n ou t:=out*cat(e,la[i]);\n od;\nout;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "PP := Par(5,3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#PPG7'7#\"\"&7$\"\"%\"\"\"7$\"\"$\"\"#7%F,F*F*7%F-F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "map(mke,PP);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%#e5G*&%#e4G\"\"\"%#e1GF'*&%#e3GF'%#e2GF'*&F*F')F(\" \"#F'*&)F+F.F'F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 129 "EmkatM(k,n) : Returns the transi tion matrix expressing the e-basis in terms of the m-basis assuming | lambda| = n and k variables" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "EmkmatM:=proc(k,n)\nlocal pars,out,cpars,rw,la,mu,te;\npars:=Par( n,k); print(pars); \ncpars:=map(conj,pars);print(cpars);\nout:=NULL; \nfor la in cpars do\n rw:=NULL;\n te:=tom(mke(la));print(te);\n fo r mu in pars do\n rw:=rw,coeff(te,m[op(mu)]);\n od;\n out:= out,[rw];\n od;\nmatrix([out]);\nend:\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "tom(e1^5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*& \"#5\"\"\"&%\"mG6$\"\"$\"\"#F&F&*&\"\"&F&&F(6$\"\"%F&F&F&&F(6#F-F&*&\" #IF&&F(6%F+F+F&F&F&*&\"#?F&&F(6%F*F&F&F&F&*&\"#gF&&F(6&F+F&F&F&F&F&*& \"$?\"F&&F(6'F&F&F&F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "EmkmatM(3,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'7#\"\"&7$\"\"% \"\"\"7$\"\"$\"\"#7%F*F(F(7%F+F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7'7'\"\"\"F%F%F%F%7&\"\"#F%F%F%7%F'F'F%7%\"\"$F%F%7$F*F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,0*&\"#5\"\"\"&%\"mG6$\"\"$\"\"#F&F&*&\"\"&F&&F( 6$\"\"%F&F&F&&F(6#F-F&*&\"#IF&&F(6%F+F+F&F&F&*&\"#?F&&F(6%F*F&F&F&F&*& \"#gF&&F(6&F+F&F&F&F&F&*&\"$?\"F&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&\"\"$\"\"\"&%\"mG6$F%\"\"#F&F&&F(6$\"\"%F&F&*&\" #7F&&F(6%F*F*F&F&F&*&\"\"(F&&F(6%F%F&F&F&F&*&\"#FF&&F(6&F*F&F&F&F&F&*& \"#gF&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,&%\"mG6 $\"\"$\"\"#\"\"\"*&\"\"&F)&F%6%F(F(F)F)F)*&F(F)&F%6%F'F)F)F)F)*&\"#7F) &F%6&F(F)F)F)F)F)*&\"#IF)&F%6'F)F)F)F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"#\"\"\"&%\"mG6%F%F%F&F&F&&F(6%\"\"$F&F&F&*&\"\" (F&&F(6&F%F&F&F&F&F&*&\"#?F&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(&%\"mG6%\"\"#F'\"\"\"F(*&\"\"$F(&F%6&F'F(F(F(F(F(*&\" #5F(&F%6'F(F(F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7'7'\"\"\"\"\"&\"#5\"#?\"#I7'\"\"!F(\"\"$\"\"(\"#77'F.F.F(\"\"#F)7'F. F.F.F(F37'F.F.F.F.F(Q)pprint186\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "EmkmatM(4,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(7# \"\"&7$\"\"%\"\"\"7$\"\"$\"\"#7%F*F(F(7%F+F+F(7&F+F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7(7'\"\"\"F%F%F%F%7&\"\"#F%F%F%7%F'F'F%7%\"\"$F% F%7$F*F'7$\"\"%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&\"#5\"\"\"&% \"mG6$\"\"$\"\"#F&F&*&\"\"&F&&F(6$\"\"%F&F&F&&F(6#F-F&*&\"#IF&&F(6%F+F +F&F&F&*&\"#?F&&F(6%F*F&F&F&F&*&\"#gF&&F(6&F+F&F&F&F&F&*&\"$?\"F&&F(6' F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&\"\"$\"\"\"&%\"m G6$F%\"\"#F&F&&F(6$\"\"%F&F&*&\"#7F&&F(6%F*F*F&F&F&*&\"\"(F&&F(6%F%F&F &F&F&*&\"#FF&&F(6&F*F&F&F&F&F&*&\"#gF&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,&%\"mG6$\"\"$\"\"#\"\"\"*&\"\"&F)&F%6%F(F(F)F )F)*&F(F)&F%6%F'F)F)F)F)*&\"#7F)&F%6&F(F)F)F)F)F)*&\"#IF)&F%6'F)F)F)F) F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"#\"\"\"&%\"mG6%F%F%F &F&F&&F(6%\"\"$F&F&F&*&\"\"(F&&F(6&F%F&F&F&F&F&*&\"#?F&&F(6'F&F&F&F&F& F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(&%\"mG6%\"\"#F'\"\"\"F(*&\" \"$F(&F%6&F'F(F(F(F(F(*&\"#5F(&F%6'F(F(F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"mG6&\"\"#\"\"\"F(F(F(*&\"\"&F(&F%6'F(F(F(F(F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7(\"\"\"\"\"&\"#5\"# ?\"#I\"#g7(\"\"!F(\"\"$\"\"(\"#7\"#F7(F/F/F(\"\"#F)F27(F/F/F/F(F5F17(F /F/F/F/F(F07(F/F/F/F/F/F(Q)pprint196\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "EE := EmkmatM(5,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7)7#\"\"&7$\"\"%\"\"\"7$\"\"$\"\"#7%F*F(F(7%F+F+F(7&F+F(F(F(7'F(F(F( F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)7'\"\"\"F%F%F%F%7&\"\"#F%F%F %7%F'F'F%7%\"\"$F%F%7$F*F'7$\"\"%F%7#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&\"#5\"\"\"&%\"mG6$\"\"$\"\"#F&F&*&\"\"&F&&F(6$\"\"% F&F&F&&F(6#F-F&*&\"#IF&&F(6%F+F+F&F&F&*&\"#?F&&F(6%F*F&F&F&F&*&\"#gF&& F(6&F+F&F&F&F&F&*&\"$?\"F&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&\"\"$\"\"\"&%\"mG6$F%\"\"#F&F&&F(6$\"\"%F&F&*&\"#7F &&F(6%F*F*F&F&F&*&\"\"(F&&F(6%F%F&F&F&F&*&\"#FF&&F(6&F*F&F&F&F&F&*&\"# gF&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,&%\"mG6$\" \"$\"\"#\"\"\"*&\"\"&F)&F%6%F(F(F)F)F)*&F(F)&F%6%F'F)F)F)F)*&\"#7F)&F% 6&F(F)F)F)F)F)*&\"#IF)&F%6'F)F)F)F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"#\"\"\"&%\"mG6%F%F%F&F&F&&F(6%\"\"$F&F&F&*&\"\" (F&&F(6&F%F&F&F&F&F&*&\"#?F&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(&%\"mG6%\"\"#F'\"\"\"F(*&\"\"$F(&F%6&F'F(F(F(F(F(*&\" #5F(&F%6'F(F(F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"mG6& \"\"#\"\"\"F(F(F(*&\"\"&F(&F%6'F(F(F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"mG6'\"\"\"F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EEGK%'matrixG6#7)7)\"\"\"\"\"&\"#5\"#?\"#I\"#g\"$?\"7)\"\"!F*\" \"$\"\"(\"#7\"#FF/7)F2F2F*\"\"#F+F5F.7)F2F2F2F*F8F4F-7)F2F2F2F2F*F3F,7 )F2F2F2F2F2F*F+7)F2F2F2F2F2F2F*Q)pprint206\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "EmkmatM(6,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7)7#\"\"&7$\"\"%\"\"\"7$\"\"$\"\"#7%F*F(F(7%F+F+F(7&F+F(F(F(7'F(F(F(F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)7'\"\"\"F%F%F%F%7&\"\"#F%F%F% 7%F'F'F%7%\"\"$F%F%7$F*F'7$\"\"%F%7#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&\"#5\"\"\"&%\"mG6$\"\"$\"\"#F&F&*&\"\"&F&&F(6$\"\"% F&F&F&&F(6#F-F&*&\"#IF&&F(6%F+F+F&F&F&*&\"#?F&&F(6%F*F&F&F&F&*&\"#gF&& F(6&F+F&F&F&F&F&*&\"$?\"F&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&\"\"$\"\"\"&%\"mG6$F%\"\"#F&F&&F(6$\"\"%F&F&*&\"#7F &&F(6%F*F*F&F&F&*&\"\"(F&&F(6%F%F&F&F&F&*&\"#FF&&F(6&F*F&F&F&F&F&*&\"# gF&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,&%\"mG6$\" \"$\"\"#\"\"\"*&\"\"&F)&F%6%F(F(F)F)F)*&F(F)&F%6%F'F)F)F)F)*&\"#7F)&F% 6&F(F)F)F)F)F)*&\"#IF)&F%6'F)F)F)F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"#\"\"\"&%\"mG6%F%F%F&F&F&&F(6%\"\"$F&F&F&*&\"\" (F&&F(6&F%F&F&F&F&F&*&\"#?F&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(&%\"mG6%\"\"#F'\"\"\"F(*&\"\"$F(&F%6&F'F(F(F(F(F(*&\" #5F(&F%6'F(F(F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"mG6& \"\"#\"\"\"F(F(F(*&\"\"&F(&F%6'F(F(F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"mG6'\"\"\"F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7)\"\"\"\"\"&\"#5\"#?\"#I\"#g\"$?\"7)\"\"!F(\"\"$\"\" (\"#7\"#FF-7)F0F0F(\"\"#F)F3F,7)F0F0F0F(F6F2F+7)F0F0F0F0F(F1F*7)F0F0F0 F0F0F(F)7)F0F0F0F0F0F0F(Q)pprint216\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "EmkmatM(7,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)7# \"\"&7$\"\"%\"\"\"7$\"\"$\"\"#7%F*F(F(7%F+F+F(7&F+F(F(F(7'F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)7'\"\"\"F%F%F%F%7&\"\"#F%F%F%7%F'F 'F%7%\"\"$F%F%7$F*F'7$\"\"%F%7#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,0*&\"#5\"\"\"&%\"mG6$\"\"$\"\"#F&F&*&\"\"&F&&F(6$\"\"%F&F&F&&F(6#F- F&*&\"#IF&&F(6%F+F+F&F&F&*&\"#?F&&F(6%F*F&F&F&F&*&\"#gF&&F(6&F+F&F&F&F &F&*&\"$?\"F&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,. *&\"\"$\"\"\"&%\"mG6$F%\"\"#F&F&&F(6$\"\"%F&F&*&\"#7F&&F(6%F*F*F&F&F&* &\"\"(F&&F(6%F%F&F&F&F&*&\"#FF&&F(6&F*F&F&F&F&F&*&\"#gF&&F(6'F&F&F&F&F &F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,&%\"mG6$\"\"$\"\"#\"\"\"*& \"\"&F)&F%6%F(F(F)F)F)*&F(F)&F%6%F'F)F)F)F)*&\"#7F)&F%6&F(F)F)F)F)F)*& \"#IF)&F%6'F)F)F)F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"# \"\"\"&%\"mG6%F%F%F&F&F&&F(6%\"\"$F&F&F&*&\"\"(F&&F(6&F%F&F&F&F&F&*&\" #?F&&F(6'F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(&%\"mG6% \"\"#F'\"\"\"F(*&\"\"$F(&F%6&F'F(F(F(F(F(*&\"#5F(&F%6'F(F(F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"mG6&\"\"#\"\"\"F(F(F(*&\"\"&F( &F%6'F(F(F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"mG6'\"\"\"F &F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7)\"\"\"\"\"& \"#5\"#?\"#I\"#g\"$?\"7)\"\"!F(\"\"$\"\"(\"#7\"#FF-7)F0F0F(\"\"#F)F3F, 7)F0F0F0F(F6F2F+7)F0F0F0F0F(F1F*7)F0F0F0F0F0F(F)7)F0F0F0F0F0F0F(Q)ppri nt226\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "inverse(EE);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7)\"\"\"!\"&\"\"&F*F)F)F *7)\"\"!F(!\"$!\"\"F*F(F)7)F,F,F(!\"#F.F*F)7)F,F,F,F(F0F.F*7)F,F,F,F,F (F-F*7)F,F,F,F,F,F(F)7)F,F,F,F,F,F,F(Q)pprint236\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "CoN := map(conj,Par(5,3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$CoNG7'7'\"\"\"F'F'F'F'7&\"\"#F'F'F'7%F)F)F'7%\" \"$F'F'7$F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ElE := map (mke, CoN);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ElEG7'*$)%#e1G\"\"& \"\"\"*&%#e2GF*)F(\"\"$F**&)F,\"\"#F*F(F**&%#e3GF*)F(F1F**&F3F*F,F*" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "map(tom,ElE);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7',0*&\"#5\"\"\"&%\"mG6$\"\"$\"\"#F'F'*&\"\"&F' &F)6$\"\"%F'F'F'&F)6#F.F'*&\"#IF'&F)6%F,F,F'F'F'*&\"#?F'&F)6%F+F'F'F'F '*&\"#gF'&F)6&F,F'F'F'F'F'*&\"$?\"F'&F)6'F'F'F'F'F'F'F',.*&F+F'F(F'F'F /F'*&\"#7F'F6F'F'*&\"\"(F'F:F'F'*&\"#FF'F>F'F'*&F=F'FBF'F',,F(F'*&F.F' F6F'F'*&F,F'F:F'F'*&FGF'F>F'F'*&F5F'FBF'F',**&F,F'F6F'F'F:F'*&FIF'F>F' F'*&F9F'FBF'F',(F6F'*&F+F'F>F'F'*&F&F'FBF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 75 "nmu(mu ) : Returns n_mu, [0*mu(1) + 1*mu(2) + 2*mu(3) + . . . + (k-1)*mu(k)] " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "nmu:=proc(mu)\nlocal i, out;\nout:=0;\nfor i from 1 to nops(mu) do\n out:=out+(i-1)*mu[i];\n od;\nout;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "nmu([ 9,4,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 101 "mk powsym(lambda) : Returns the power symmetric function p_lambda(1) * p_ lambda(2) * ... * p_lambda(k)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "mkpowsym:=proc(lambda)\nlocal i,out;\nout:=1;\nfor i in lambda \+ do\n out:=out*cat(p,i);\n od;\nout;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "mkpowsym([3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#p3G\"\"\"%#p2GF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Permutation procedures" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 114 "invsig(sig) : Returns the inverse of the permutation \+ sig (sig is written as the bottom line in 2-line notatation)." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "invsig:=proc(sig)\nlocal ou t,i,n;\nn:=nops(sig);\nout:=[seq(0,i=1..n)];\nfor i from 1 to n do\n \+ out[sig[i]]:=i;\n od;\nout;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sig:=invsig([2,1,4,6,5,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sigG7(\"\"#\"\"\"\"\"'\"\"$\"\"&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "invsig(sig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"#\"\"\"\"\"%\"\"'\"\"&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "sgn (sig) : Returns the sign of the permutation sig" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "sgn := proc (sig) " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "local i, j, n, out;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "n := nops(sig); out := 1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "for i to n-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " for j fro m i+1 to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " if sig[j] < \+ sig[i] then out := -out fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " \+ od" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "out" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sgn([1,3,2,4,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 63 "cycstr(sig) : Return s the cycle structure of sig as a partition" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "cycstr := proc (sig) " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "local out, i, j, cyc, S, a, b, mu;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "S := \{op(sig)\}; out := NULL;" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "while 0 < nops(S) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " cyc := NULL; i := S[1]; a := i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " while b <> i do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " b := sig[a];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " \+ cyc := cyc, b;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " a := b;" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " S := `minus`(S,\{a\})" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " out := out, [cyc]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "mu := par(map(nops,[ out]))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sig:=[3,4,7,2,5,6,1]:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "cycstr(sig);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7&\"\"$\"\"#\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sg n(sig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 90 "xsi g(sig) : Returns x_sig; this is how we represent sig as an element of \+ the group algebra" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "xsig:=s ig->x[op(sig)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xsigGf*6#%$sigG6 \"6$%)operatorG%&arrowGF(&%\"xG6#-%#opG6#9$F(F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "xs:=xsig([3,2,1]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xsG&%\"xG6%\"\"$\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "op(xs);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$ \"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "ixsig(xsigma) : Returns sig" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ixsig:=var->[op(var)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ixsigGf*6#%$varG6\"6$%)operatorG%&arrowGF(7# -%#opG6#9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "xs:=xsi g([3,2,1]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xsG&%\"xG6%\"\"$ \"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ixsig(xs);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"$\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 73 "nlocit(sig) : Returns the location of a permutation in the canonical \+ list" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "nlocit:=proc(sig)\n local i,pers,n,out,x;\noptions remember;\nn:=nops(sig);\npers:=permute ([seq(i,i=1..n)]); \nfor i from 1 to nops(pers) do\n x[op(pers[i ])]:=i;\n od;\nout:=x[op(sig)];\nend: \n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "map(nlocit,permute([1,2,3,4,5]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7dr\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\" \"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B \"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\" #T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"# e\"#f\"#g\"#h\"#i\"#j\"#k\"#l\"#m\"#n\"#o\"#p\"#q\"#r\"#s\"#t\"#u\"#v \"#w\"#x\"#y\"#z\"#!)\"#\")\"##)\"#$)\"#%)\"#&)\"#')\"#()\"#))\"#*)\"# !*\"#\"*\"##*\"#$*\"#%*\"#&*\"#'*\"#(*\"#)*\"#**\"$+\"\"$,\"\"$-\"\"$. \"\"$/\"\"$0\"\"$1\"\"$2\"\"$3\"\"$4\"\"$5\"\"$6\"\"$7\"\"$8\"\"$9\"\" $:\"\"$;\"\"$<\"\"$=\"\"$>\"\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "nlocit([5,3,7,2,6,1,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%FK" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "SG3 :=permute([1,2,3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map( nlocit,SG3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"\"\"#\"\"$\"\" %\"\"&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 103 "mulper(alpha,beta) : Returns the permu tation in two-line form of the multiplication of two permutations" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "mulper:=proc(alpha,beta)\nl ocal n,i,out,a,b;\nn:=nops(alpha);\nout:=NULL;\nfor i from 1 to n do\n a:=beta[i];\n b:=alpha[a];\n out:=out,b;\n od;\n[out];\nend:\n \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "mulper([3,1,5,2,4],[2,5 ,4,3,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"\"\"\"%\"\"#\"\"&\" \"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "nlocit(mulper([3,1,5 ,2,4],[2,5,4,3,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 88 "Gdet(n) : Returns the matrix associated to the group de terminant for symmetric group S_n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "Gdet:=proc(n)\nlocal pers,out,i,alpha,beta,ibeta,rw, place,gamma;\npers:=permute([seq(i,i=1..n)]);\nout:=NULL;\nfor alpha i n pers do\n rw:=NULL;\n for beta in pers do\n ibeta:=invsig(beta); \n gamma:=mulper(alpha,ibeta);\n place:=nlocit(gamma);\n rw:=rw, cat(x,place);\n od;\n out:=out,[rw];\n od;\narray([out]);\nend:\n \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "GG3 := Gdet(3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GG3GK%'matrixG6#7(7(%#x1G%#x2G%#x3G %#x5G%#x4G%#x6G7(F+F*F-F,F/F.7(F,F.F*F/F+F-7(F.F,F/F*F-F+7(F-F/F+F.F*F ,7(F/F-F.F+F,F*Q)pprint246\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "factor(det(GG3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,.%#x2G \"\"\"%#x4G!\"\"%#x6GF'%#x1GF)%#x5GF)%#x3GF'F',.F&F'F-F'F(F'F*F'F+F'F, F'F'),:*$)F-\"\"#F'F'*&F-F'F&F'F)*$)F&F3F'F'*&F*F'F-F'F)*&F*F'F&F'F)*$ )F*F3F'F'*$)F+F3F'F)*&F(F'F+F'F'*$)F(F3F'F)*&F,F'F+F'F'*&F,F'F(F'F'*$) F,F3F'F)F3F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 79 "sigact(sig,P) : Returns the objec t sig(P) where P is an object using x1, ... xn" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "sigact:=proc(sig,P)\nlocal i,pat,out;\npat:=\{s eq( cat(x,i)=cat(x,sig[i]),i=1..nops(sig))\};\nout:=subs(pat,P);\nend ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigactGf*6$%$sigG%\"PG6%%\"iG% $patG%$outG6\"F-C$>8%<#-%$seqG6$/-%$catG6$%\"xG8$-F76$F9&9$6#F:/F:;\" \"\"-%%nopsG6#F>>8&-%%subsG6$F09%F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sigact([4,1,3,2],proc1(x1,x2)*proc2(x3,x4));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%&proc1G6$%#x4G%#x1G\"\"\"-%&proc2G 6$%#x3G%#x2GF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eg := sig act([4,1,3,2],\{x1,x2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#egG<$% #x1G%#x4G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sigact([4,1,3, 2],eg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$%#x2G%#x4G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "mkord(k,n) : Returns Ordered k-tuples of numbers from 1 to n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "mkord:=proc(k,n)\nlocal i, out,F,pers;\npers:=permute([seq(i,i=1..n)]);\nF:=Y[seq(cat(x,i),i=1..k )];\nout:=[op(\{op(map(sigact,pers,F))\})];\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "LL := mkord(2,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#LLG7.&%\"YG6$%#x1G%#x3G&F'6$F)%#x2G&F'6$F-F*&F'6$%#x 4GF)&F'6$F2F-&F'6$F*F-&F'6$F*F2&F'6$F-F2&F'6$F)F2&F'6$F-F)&F'6$F2F*&F' 6$F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "map2(sigact,[2,1, 4,3],mkord(1,4) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%\"YG6#%#x2G& F%6#%#x1G&F%6#%#x4G&F%6#%#x3G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 119 "knordact(sig, k, n) : Returns k-tuples of 1, ..., n, then acts on it by sig and outputs cor responding pemutation matrix" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "knordact:=proc(sig,k,n)\nlocal N,out,rw,mat,bas,imbas,S,T;\nbas:= mkord(k,n);print(bas);\nimbas:=map2(sigact,sig,bas);print(imbas);\nN:= nops(bas);print(N);\nmat:=NULL;\nfor S in imbas do\n rw:=NULL;\n for T in bas do\n if T=S then rw:=rw,1; else rw:=rw,0; fi;\n \+ od;\n mat:=mat,[rw];\n od;\ntranspose(matrix([mat]));\nend: \n\n\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "knordact([3,2,1],2,3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7(&%\"YG6$%#x1G%#x3G&F%6$F'%#x2G&F%6$F +F(&F%6$F(F+&F%6$F+F'&F%6$F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(&% \"YG6$%#x3G%#x1G&F%6$F'%#x2G&F%6$F+F(&F%6$F(F+&F%6$F+F'&F%6$F(F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7(\"\"!F(F(F(F(\"\"\"7(F(F(F(F)F(F(7(F(F(F(F(F)F(7(F( F)F(F(F(F(7(F(F(F)F(F(F(7(F)F(F(F(F(F(Q)pprint256\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 78 "descmon(sig) : Returns the descent monomial associated to sig (WHAT \+ IS THIS?)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "descmon:=proc( sig)\nlocal i,out,seg,n;\nn:=nops(sig);\nout:=1;\nseg:=1;\nfor i from \+ 1 to n-1 do\n seg:=seg*x[sig[i]];\n if sig[i]>sig[i+1] th en\n out:=out*seg;\n fi;\n od;\nout;\ne nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "permute(3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7(7%\"\"\"\"\"#\"\"$7%F%F'F&7%F&F%F'7% F&F'F%7%F'F%F&7%F'F&F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "D es3:=map(descmon,permute(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%De s3G7(\"\"\"*&&%\"xG6#F&F&&F)6#\"\"$F&&F)6#\"\"#*&F.F&F+F&F+*&F.F&)F+F0 F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "descmon([1,7,4,5,2,6, 3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.)&%\"xG6#\"\"\"\"\"$F()&F&6# \"\"(F)F()&F&6#\"\"%\"\"#F()&F&6#\"\"&F2F(&F&6#F2F(&F&6#\"\"'F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Tableau p rocedures" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "shape(tab) : Returns the shape of a tableau as a partition" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "shape := proc (tab) options operator, arrow; " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "map(nops,tab)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tab o:=[[1,2,3],[4,5],[6],[7]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "shap e(tabo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"$\"\"#\"\"\"F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 66 "disp(tab) : Returns a very nice visual representation o f a tableau" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "mkzs := proc \+ (m) local i; `$`(` `,i = 1 .. m) end: # This is just a helper procedu re\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "disp := proc (tab) \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "local sh, i, out, k, mx, row; \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "sh := shape(tab); k := nops(tab ); mx := max(op(sh)); out := NULL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "for i to k do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " row := [mkzs (1), op(tab[k+1-i]), mkzs(mx+1-sh[k+1-i])];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " out := out, row" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "array([out])" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tabo:=[[1,2,3],[4,5],[6],[7]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "disp(tabo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'% \"~G\"\"(F(F(F(7'F(\"\"'F(F(F(7'F(\"\"%\"\"&F(F(7'F(\"\"\"\"\"#\"\"$F( Q)pprint266\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 97 "addo(i,tab,s) : Returns a tableau with the letter `s` added to row `i` of tab, a standard tableau" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "addo := proc (i, tab, s) \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "local k, out, te, j;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "k := nops(tab); out := NULL;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "if k < i then out := ob(tab), [s]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "else for j to i-1 do out := out, \+ tab[j] od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "te := ob(tab[i]), s; \+ out := out, [te];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "for j from i+1 to k do out := out, tab[j] od" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "fi ; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "[out]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tab o:=[[1,2,3],[4,5],[6],[7]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "disp(tabo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'%\" ~G\"\"(F(F(F(7'F(\"\"'F(F(F(7'F(\"\"%\"\"&F(F(7'F(\"\"\"\"\"#\"\"$F(Q) pprint276\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "disp(addo(3, tabo,8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'%\"~G\"\" (F(F(F(7'F(\"\"'\"\")F(F(7'F(\"\"%\"\"&F(F(7'F(\"\"\"\"\"#\"\"$F(Q)ppr int286\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 71 "prog(tab) : Returns a list of all success ors of tab, a standard tableau" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "prog:=proc(tab)\nlocal out,cors,sh,i,n,k;\nout:=NULL;\nsh:=shap e(tab);\nn:=size(sh);\ncors:=corns(sh);\nk:=nops(cors);\nfor i from 1 \+ to k do\n out:=out,addo(cors[k+1-i][1],tab,n+1);\n od;\n[ out];\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tabo:=[[1, 2,3],[4,5],[6],[7]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dis p(tabo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'%\"~G\"\"( F(F(F(7'F(\"\"'F(F(F(7'F(\"\"%\"\"&F(F(7'F(\"\"\"\"\"#\"\"$F(Q)pprint2 96\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "map(disp,prog(tabo )); # map(f,list) is a Maple function which returns a list containing f(list[i]) for all i." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&K%'matrixG 6#7'7'%\"~G\"\")F)F)F)7'F)\"\"(F)F)F)7'F)\"\"'F)F)F)7'F)\"\"%\"\"&F)F) 7'F)\"\"\"\"\"#\"\"$F)Q)pprint306\"KF%6#7&F+7'F)F.F*F)F)F/F2Q)pprint31 F7KF%6#7&F+F-7'F)F0F1F*F)F2Q)pprint32F7KF%6#7&7(F)F,F)F)F)F)7(F)F.F)F) F)F)7(F)F0F1F)F)F)7(F)F3F4F5F*F)Q)pprint33F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 93 "maddo( tablist) : Returns a list of all successors of all tableaux in tablist , without repeats" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "maddo: =proc(sectab)\nlocal i,out,N;\nN:=nops(sectab);\nout:=NULL;\nfor i fro m 1 to N do\n out:=out,ob(prog(sectab[i]));\n od;\n[out];\nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ST2:=[[[1],[2]],[[1,2]] ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "map(disp,ST2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$K%'matrixG6#7$7%%\"~G\"\"#F)7%F)\"\" \"F)Q)pprint346\"KF%6#7#7&F)F,F*F)Q)pprint35F." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "ST3:=maddo(ST2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "map(disp,ST3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&K %'matrixG6#7%7%%\"~G\"\"$F)7%F)\"\"#F)7%F)\"\"\"F)Q)pprint366\"KF%6#7$ 7&F)F,F)F)7&F)F.F*F)Q)pprint37F0KF%6#7$7&F)F*F)F)7&F)F.F,F)Q)pprint38F 0KF%6#7#7'F)F.F,F*F)Q)pprint39F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 137 "mkstad(n) \+ : Returns a list of all standard tableaux of size n \n(ST2, ST3, ST4, \+ ST5, ST6 are the sets of standard tableau of those orders)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "mkstad:=proc(n)\nlocal i,te,tabs; \nte:=maddo([[[1]]]);\nfor i from 1 to n-2 do\n te:=maddo(te);\n od ;\ntabs:=te;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ST2:= mkstad(2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ST3:=mkstad(3 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ST4:=mkstad(4):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ST5:=mkstad(5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ST6:=mkstad(6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "map(disp,ST5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7 " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 79 "pck(tablist, la) : R eturns a list of every tableau in tablist with shape lambda" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "pck:=proc(sectab,sha)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "local out,i,shtb;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "out:=NULL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "for \+ i from 1 to nops(sectab) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " s htb:=shape(sectab[i]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " if equ a(shtb,sha) then out:=out,sectab[i];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6 "[out];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "map(disp,pck(ST5,[3,2] ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'K%'matrixG6#7$7'%\"~G\"\"#\" \"%F)F)7'F)\"\"\"\"\"$\"\"&F)Q)pprint666\"KF%6#7$7'F)F*F/F)F)7'F)F-F.F +F)Q)pprint67F1KF%6#7$7'F)F.F+F)F)7'F)F-F*F/F)Q)pprint68F1KF%6#7$7'F)F .F/F)F)7'F)F-F*F+F)Q)pprint69F1KF%6#7$7'F)F+F/F)F)7'F)F-F*F.F)Q)pprint 70F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 80 "fillo(tab) : Returns a list of equal sized part s by filling a tableau with zeros" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fillo := proc (tab) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "local h, k, out, i, j, sh, te;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sh := shape(tab); h := sh[1]; k := nops(sh); out := N ULL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "for i to k do " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 18 " te := NULL; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " for j to sh[i] do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " te := te, tab[i][j]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " \+ od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " for j from sh[i]+1 t o h do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " te := te, 0" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " out := out, [te]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "[out]" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tabo:=[[1,2,3],[4,5],[6],[7]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fillo(tabo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7% \"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"!7%\"\"'F+F+7%\"\"(F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "disp(fillo(tabo));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'%\"~G\"\"(\"\"!F*F(7'F(\"\"'F*F*F( 7'F(\"\"%\"\"&F*F(7'F(\"\"\"\"\"#\"\"$F(Q)pprint716\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "trans(tab) : Returns the transpose of a standard tableau" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "trans := proc (tab) \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "local mat, sh, csh, out, tmat , te, i, j;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "sh := shape(tab); cs h := conj(sh); mat := array(fillo(tab)); tmat := transpose(mat); out : = NULL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i to nops(csh) do" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " te := NULL;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 21 " for j to csh[i] do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " te := te, tmat[i,j]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " out := out, [te]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 5 "[out]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "en d:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tabo:=[[1,2,3],[4,5], [6],[7]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "disp(tabo);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'%\"~G\"\"(F(F(F(7'F( \"\"'F(F(F(7'F(\"\"%\"\"&F(F(7'F(\"\"\"\"\"#\"\"$F(Q)pprint726\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "disp(trans(tabo));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7(%\"~G\"\"$F(F(F(F(7(F(\"\"# \"\"&F(F(F(7(F(\"\"\"\"\"%\"\"'\"\"(F(Q)pprint736\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 100 "inter(tab1,tab2) : Returns the intersection of two injective tabl eaux in a format 'disp' can display" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "inter := proc (T1, T2) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "local sh, i, j, out, tr2, k, h, te, ints;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "sh := shape(T1); tr2 := trans(T2); \+ k := nops(T1); h := nops(tr2); out := NULL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "for i to k do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " \+ te := NULL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " for j to sh[i] \+ do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " ints := `intersect`(\{o b(T1[i])\},\{ob(tr2[j])\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " \+ te := te, ints" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " od;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " out := out, [te] " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "[ou t]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ST311:=pck(ST5,[3,1,1]): # ST311 is all std. tab leaux with shape 3,1,1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(disp,ST311);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7(K%'matrixG6#7%7'%\"~G\"\"$F)F)F)7'F) \"\"#F)F)F)7'F)\"\"\"\"\"%\"\"&F)Q)pprint746\"KF%6#7%7'F)F/F)F)F)F+7'F )F.F*F0F)Q)pprint75F2KF%6#7%7'F)F0F)F)F)F+7'F)F.F*F/F)Q)pprint76F2KF%6 #7%F6F(7'F)F.F,F0F)Q)pprint77F2KF%6#7%F " 0 "" {MPLTEXT 1 0 31 "disp(inter(ST311[1],ST311[2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7'%\"~G<\"F(F(F(7'F(<#\"\"#F(F(F(7'F(<$\"\"\"\"\"% F)<#\"\"&F(Q)pprint806\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "intlist:=map(inter,ST311,ST311[2]): # intlist is a list with the \+ intersection of every 3,1,1 tableaux with ST311[2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "map(disp,intlist);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(K%'matrixG 6#7%7'%\"~G<\"F)F)F)7'F)<#\"\"#F)F)F)7'F)<$\"\"\"\"\"%F*<#\"\"&F)Q)ppr int816\"KF%6#7%7'F)<#F1F)F)F)F+7'F)<#F0<#\"\"$F2F)Q)pprint82F5KF%6#7%F (F+7'F)F/F=F*F)Q)pprint83F5KF%6#7%F9F(7'F)<$F0F-F*F2F)Q)pprint84F5KF%6 #7%F(F(7'F)<%F0F-F1F*F*F)Q)pprint85F5KF%6#7%F(F97'F)FIF=F*F)Q)pprint86 F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "act(sig,tab) : Returns the tableau formed by sig \+ acting on tab" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "act:=proc( sig,tab)\nlocal sh,i,j,out,te;\nsh:=shape(tab);\nout:=NULL;\nfor i fro m 1 to nops(sh) do\n te:=NULL;\n for j from 1 to sh[i] do\n te: =te,sig[tab[i][j]];\n od;\n out:=out,[te];\n od;\n[out];\nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ST32:=pck(ST5,[3,2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "T:=ST32[5]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "disp(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7'%\"~G\"\"%\"\"&F(F(7'F(\"\"\"\"\"#\"\"$ F(Q)pprint876\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sig:=[3, 4,2,5,1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "disp(act(sig,T ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7'%\"~G\"\"&\"\" \"F(F(7'F(\"\"$\"\"%\"\"#F(Q)pprint886\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 99 "secact(sig, tablist) : Returns a list of tableaux formed by acting on each tableau of tablist by sig" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "secact :=(sig,L)->map2(act,sig,L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'seca ctGf*6$%$sigG%\"LG6\"6$%)operatorG%&arrowGF)-%%map2G6%%$actG9$9%F)F)F) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sig:=[3,4,2,5,1]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ST32:=pck(ST5,[3,2]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "map(disp,ST32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'K%'matrixG6#7$7'%\"~G\"\"#\"\"%F)F)7'F)\"\"\" \"\"$\"\"&F)Q)pprint896\"KF%6#7$7'F)F*F/F)F)7'F)F-F.F+F)Q)pprint90F1KF %6#7$7'F)F.F+F)F)7'F)F-F*F/F)Q)pprint91F1KF%6#7$7'F)F.F/F)F)7'F)F-F*F+ F)Q)pprint92F1KF%6#7$7'F)F+F/F)F)7'F)F-F*F.F)Q)pprint93F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "map(disp,secact(sig,ST32));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7'K%'matrixG6#7$7'%\"~G\"\"%\"\"&F)F)7 'F)\"\"$\"\"#\"\"\"F)Q)pprint946\"KF%6#7$7'F)F*F/F)F)7'F)F-F.F+F)Q)ppr int95F1KF%6#7$7'F)F.F+F)F)7'F)F-F*F/F)Q)pprint96F1KF%6#7$7'F)F.F/F)F)7 'F)F-F*F+F)Q)pprint97F1KF%6#7$7'F)F+F/F)F)7'F)F-F*F.F)Q)pprint98F1" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Auxiliar y Polynomial procedures" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "prxi(n ) : Returns the monomial x1*x2*...*xn" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "prxi:=n->convert([seq(x[i],i=1..n)],`*`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%prxiGf*6#%\"nG6\"6$%)operatorG%&arrowGF(- %(convertG6$7#-%$seqG6$&%\"xG6#%\"iG/F6;\"\"\"9$%\"*GF(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "prxi(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**&%\"xG6#\"\"\"F'&F%6#\"\"#F'&F%6#\"\"$F'&F%6#\"\"%F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 88 "art(n) : Returns the artin monomials in n vars (r art(n) retruns assuming reverse order)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "art:=proc(n)\nlocal i,j,te,out,sm;\nout:=1;\nfor i f rom 2 to n do\n sm:=convert([seq(x[i]^(j),j=0..i-1)],`+`);\n out:=out *sm;\n od;\n[op(expand(out))];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "art(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"&% \"xG6#\"\"#&F&6#\"\"$*&F%F$F)F$*$)F)F(F$*&F%F$F.F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "rart:=proc(n)\nlocal i,j,te,out,sm;\nout:= 1;\nfor i from 1 to n-1 do\n sm:=convert([seq(x[i]^(n-i-j),j=0..n-i)] ,`+`);\n out:=out*sm;\n od;\n[op(expand(out))];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rart(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"*$)&%\"xG6#F$\"\"#F$F'&F(6#F**&F+F$F&F$*&F+F$F' F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 164 "getexpo(mon,n) : Returns the exponent sequence o f monomial mon assuming n vars\ngetcomp(mon,n) : Returns the exponent \+ sequence as composition (i.e. eliminates zeros)" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 112 "getexpo:=proc(mon,n)\nlocal i,out;\nout:=NULL ;\nfor i from 1 to n do\n out:=out,degree(mon,x[i]);\n od;\n[out];\n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "getexpo(x[1]^2*x[2] ,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"#\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "getcomp:=proc(mon,n)\nlocal i,out; \nout:=NULL;\nfor i from 1 to n do\nif degree(mon,x[i])>0 then\n out:= out,degree(mon,x[i]);\n fi;\n od;\n[out];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "getcomp(x[1]*x[3]^2,4);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 137 "schurf(composition) : Returns schur function corresponding to the DELTA of composition wi th distinct parts\nas if computed via slinky rule" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "schurf:=proc(p)\nlocal i,out,sg,siz,sp,d,la, n;\nn:=nops(p); \nsiz:=size(p);\nif siz " 0 "" {MPLTEXT 1 0 18 "schurf([3,8,5,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"sG6&\"\"&\"\"$\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 110 "monP(mon,P,n) : Retruns the scalar product of monomial mon wit h polynomial P assuming n vars as schur function" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 374 "monP:=proc(mon,P,n)\nlocal out,moes,coes,al,i ,mons,sc,a,b;\nal:=[seq(x[i],i=1..n)];\ncoes:=[coeffs(expand(P),al,`mo es`)];\nmons:=[moes];\na:=getexpo(mon,n);\nout:=0;\nfor i from 1 to no ps(mons) do\nb:=getexpo(mons[i],n);#print(i,b);\n\n sc:=schurf(a+b); \n#if sc<>0 then print(b,` -> `,sc,` ---> `,a+b,` -----> `,coes[i]) ;fi;\n\n out:=out+coes[i]*sc;\n od;\nsubs(s[]=1,tos(out)); \nend:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "monP(x[1]*x[2]^2,x[2]*x[3]^ 2,3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$&%\"sG6%\"\"\"F'F'!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "monP(x[1]*x[2]^2,x[3] ,3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 82 "scal PQ(P,Q,n) : Returns the scalar product of two polynomials P, Q assumin g n vars" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "scalPQ:=proc(P, Q,n)\nlocal out,moes,coes,al,i,mon,coe,sc,buffo;\nal:=[seq(x[i],i=1..n )];\ncoes:=[coeffs(expand(P),al,`moes`)];\nbuffo:=[moes];\nout:=0;\nfo r i from 1 to nops(coes) do\n coe:=coes[i];\n mon:=buffo[i]; \n \+ sc:=monP(mon,Q,n);\n out:=out+coe*sc;\n od;\nout;\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "scalPQ(art(3)[6],art(3)[5],3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"sG6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "art(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7(\"\"\"&%\"xG6#\"\"#&F&6#\"\"$*&F%F$F)F$*$)F)F(F$*&F%F$F.F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "expand(evalsf(s[2],x1+x2+x3) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%#x1G\"\"#\"\"\"F(*$)%#x2GF 'F(F(*$)%#x3GF'F(F(*&F+F(F&F(F(*&F.F(F&F(F(*&F.F(F+F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "mylex(A,B) : Tests if A < B in degree lex " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 404 "mylex:=proc(A,B)\nlocal out,i,a,b,bool,sa,sb ;\na:=nops(A);\nb:=nops(B);\nsa:=convert(A,`+`);\nsb:=convert(B,`+`); \nif sa>sb then out:=false;\n elif sab \+ then out:=false;\n elif aB[i] then\n out:=true;bool:=false;\n fi;\n od;\nfi;\nout;\nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "mylex(x[3],x[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "dart(n) : Returns artin monomials in dlex order as compositions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 " dart:=proc(n)\nlocal arts,exarts;\narts:=[op(art(n))];\nexarts:=map(ge texpo,arts,n);\nsort(exarts,mylex);\nend:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "dart(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7:7&\"\" !F%F%F%7&F%\"\"\"F%F%7&F%F%F'F%7&F%F%F%F'7&F%F'F'F%7&F%F'F%F'7&F%F%\" \"#F%7&F%F%F'F'7&F%F%F%F-7&F%F'F-F%7&F%F'F'F'7&F%F'F%F-7&F%F%F-F'7&F%F %F'F-7&F%F%F%\"\"$7&F%F'F-F'7&F%F'F'F-7&F%F'F%F67&F%F%F-F-7&F%F%F'F67& F%F'F-F-7&F%F'F'F67&F%F%F-F67&F%F'F-F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Ar := [op(art(4))];" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#ArG7:\"\"\"&%\"xG6#\"\"#&F(6#\"\"$*&F'F&F+F&*$)F+F*F&*&F'F&F0F&&F (6#\"\"%*&F2F&F'F&*&F2F&F+F&*(F2F&F+F&F'F&*&F2F&F0F&*(F2F&F0F&F'F&*$)F 2F*F&*&F;F&F'F&*&F;F&F+F&*(F;F&F+F&F'F&*&F;F&F0F&*(F;F&F0F&F'F&*$)F2F- F&*&FBF&F'F&*&FBF&F+F&*(FBF&F+F&F'F&*&FBF&F0F&*(FBF&F0F&F'F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "map(getexpo,Ar,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7:7&\"\"!F%F%F%7&F%\"\"\"F%F%7&F%F%F'F%7&F%F 'F'F%7&F%F%\"\"#F%7&F%F'F+F%7&F%F%F%F'7&F%F'F%F'7&F%F%F'F'7&F%F'F'F'7& F%F%F+F'7&F%F'F+F'7&F%F%F%F+7&F%F'F%F+7&F%F%F'F+7&F%F'F'F+7&F%F%F+F+7& F%F'F+F+7&F%F%F%\"\"$7&F%F'F%F:7&F%F%F'F:7&F%F'F'F:7&F%F%F+F:7&F%F'F+F :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 97 "mkmon(list) : Returns the monomial from the given \+ exponent sequence (mkmon2 does similar command)" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 97 "mkmon:=proc(p)\nlocal i,out;\nout:=1;\nfor i f rom 1 to nops(p) do\n out:=out*x[i]^p[i];\nod;\nout;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "mkmon([3,2,1]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*()&%\"xG6#\"\"\"\"\"$F()&F&6#\"\"#F-F(&F&6#F)F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "mkmon2:=proc(p)\nlocal i,ou t;\nout:=1;\nfor i from 1 to nops(p) do\n out:=out*(cat(x,i))^p[i];\no d;\nout;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "mkmon([1, 2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"xG6#\"\"\"F')&F%6#\"\"#F +F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "mkmon2([1,2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#x1G\"\"\")%#x2G\"\"#F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "# Notice that mkmon and mkmon2 are similar but create the monomials as x[1]*x[2]^2 or x1*x2^2 accordingl y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 98 "LD(P ,n) : Returns coefficients and exponent sequence of leading monomial o f polynomial P in n vars" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "LD:=proc(P,n)\nlocal i,out,al,mon,d,coe,pr;\nal:=[seq(cat(x,i),i=1..n )];\npr:=leadmon(expand(P),al,plex);\ncoe:=pr[1];\nmon:=pr[2];\nout:=N ULL;\nfor i from 1 to n do\n d:=degree(mon,cat(x,i));\n if d>0 then out:=out,d; fi; \n od;\n[coe,[out]];\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "LD(evalsf(m[3,2,2],x1+x2+x3),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"7%\"\"$\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sort(expand(evalsf(m[3,2,2],x1+x2+x3)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*()%#x3G\"\"$\"\"\")%#x2G\"\"#F()%#x 1GF+F(F(*()F&F+F()F*F'F(F,F(F(*(F/F(F)F()F-F'F(F(" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 127 "lead(P,n) : Returns coefficient and leading mo nomial assuming plex ordering of variables\nleadmon is more general ma ple command " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "lead:=proc(P ,n)\nlocal vars,out,i;\nvars:=[seq(x[n+1-i],i=1..n)];\nout:=leadmon(P, vars,plex);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "lead(x [1]^2+2*x[3]^4+3*x[1]*x[3]^3,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\" \"#*$)&%\"xG6#\"\"$\"\"%\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(x[3+1-i],i=1..3);x[3], x[2], x[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%&%\"xG6#\"\"$&F$6#\"\"#&F$6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%&%\"xG6#\"\"$&F$6#\"\"#&F$6#\"\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 60 "# \"leadmon\" is built-in maple command, see ?leadmon for help" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 306 "testAI(P,BB,n) : Returns polynom ial P written in terms of basis elements (BB[i] becomes y[i]) and the \+ elements x1,..., xn\nIf P is in the ideal of , then P will be writ ten only in terms of y's \nFor it to work properly, n should equal num ber of variables and BB must be a true linearly independent basis" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "testAI:=proc(P,BB,n)\nlocal i,N,out,bas,al,gbas;\nN:=nops(BB);\nal:=[seq(cat(x,i),i=1..n),seq(cat (y,i),i=1..n)];\nbas:=[seq(BB[i]-cat(y,i),i=1..N)];\ngbas:=gbasis(bas, al, plex);\nout:=normalf(P,gbas,al);\nend:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "P1:=evalsf(p1,x1+x2+x3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G,(%#x1G\"\"\"%#x2GF'%#x3GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P2:=evalsf(p2,x1+x2+x3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P2G,(*$)%#x1G\"\"#\"\"\"F**$)%#x2GF)F*F**$)%#x3GF)F* F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P3:=evalsf(p3,x1+x2+x 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P3G,(*$)%#x1G\"\"$\"\"\"F**$ )%#x2GF)F*F**$)%#x3GF)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Pol4:=evalsf(p4,x1+x2+x3)+x1^2*x2^2+x1^2*x3^2+x2^2*x3^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Pol4G,.*$)%#x1G\"\"%\"\"\"F**$)%#x2GF)F*F **$)%#x3GF)F*F**&)F-\"\"#F*)F(F3F*F**&)F0F3F*F4F*F**&F2F*F6F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "testAI(Pol4,[P1,P2,P3],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"#7F&*$)%#y1G\"\"%F&F&F& *&#F&\"\"#F&*&%#y2GF&)F*F.F&F&!\"\"*&#\"\"$F+F&*$)F0F.F&F&F&*&#F.F5F&* &F*F&%#y3GF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "top(sub s(e4=0,toe(p4)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"'F &*$)%#p1G\"\"%F&F&F&*&%#p2GF&)F*\"\"#F&!\"\"*&#F+\"\"$F&*&%#p3GF&F*F&F &F&*&#F&F/F&*$)F-F/F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "GB:=gbasis([P1,P2,P3],[x1,x2,x3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GBG7%*$)%#x3G\"\"$\"\"\",(*$)%#x2G\"\"#F*F**&F(F*F.F*F**$)F(F /F*F*,(%#x1GF*F.F*F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 " normalf(x2^2+x3^2+x2*x3,GB,[x1,x2,x3]); # = 0 implies that polynomial is in the ideal of P1, P2, P3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 74 "# gbasis is built in maple command that makes \+ gbasis, see ?gbasis for help" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 58 "Construction of matrices in Young's natur al representation" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 112 "per(T1,T2) \+ : When T1 ^ T2 is good, returns permutation beta in the col space of T 2, such that T2 = beta(T1) ^ T2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "per:=proc(T1,T2)\nlocal i,j,sig,sh,k,n,lsig,out;\nsh:=shape(T 1);\nk:=nops(sh);\nsig:=NULL;\nfor i from 1 to k do\n for j from 1 to sh[i] do\n sig:=sig,[T1[i][j],T2[i][j]];\n \+ od;\n od;\nlsig:=sort([sig],lexo);\nn:=nops(lsig);\nout: =NULL;\nfor i from 1 to n do out:=out,lsig[i][2]; od;\n[out]\nend:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ST32:=pck(ST5,[3,2]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "map(disp,ST32); # All std. \+ tableaux of shape 3,2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'K%'matrixG6 #7$7'%\"~G\"\"#\"\"%F)F)7'F)\"\"\"\"\"$\"\"&F)Q)pprint996\"KF%6#7$7'F) F*F/F)F)7'F)F-F.F+F)Q*pprint100F1KF%6#7$7'F)F.F+F)F)7'F)F-F*F/F)Q*ppri nt101F1KF%6#7$7'F)F.F/F)F)7'F)F-F*F+F)Q*pprint102F1KF%6#7$7'F)F+F/F)F) 7'F)F-F*F.F)Q*pprint103F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "T1:=ST32[1]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "T5:=ST32[5]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "disp(inter(T1,T5)); # We first ch eck that T1 ^ T2 is good" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7$7'%\"~G<#\"\"%<#\"\"#F(F(7'F(<#\"\"\"<#\"\"&<#\"\"$F(Q*pprint1046 \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "beta:=per(T1,T5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG7'\"\"\"\"\"%\"\"#\"\"&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sgn(beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 70 "CC(T1,T2) : Returns 0 if (T1^T2) is bad , sign(beta) if (T1^T2) is good" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "CC := proc (A, B) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "local bool, sh, i, j, tetab, itab, te, k, sss, beta, \+ out;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "option remember; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sh := shape(A); k := nops(sh); tetab := i nter(A,B); bool := true; itab := NULL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for i to k while bool do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " te := NULL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " for j to sh[i ] do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " sss := [ob(tetab[i][j ])];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " if nops(sss) <> 1 the n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " out := 0; bool := fal se" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " else te := te, ob(ss s)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " \+ itab := itab, [te]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "if bool then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " beta := per([itab],B);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " out := sgn(beta)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "out" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ST32:=pck(ST5,[3,2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "map(disp,ST32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'K%'matrixG6#7 $7'%\"~G\"\"#\"\"%F)F)7'F)\"\"\"\"\"$\"\"&F)Q*pprint1056\"KF%6#7$7'F)F *F/F)F)7'F)F-F.F+F)Q*pprint106F1KF%6#7$7'F)F.F+F)F)7'F)F-F*F/F)Q*pprin t107F1KF%6#7$7'F)F.F/F)F)7'F)F-F*F+F)Q*pprint108F1KF%6#7$7'F)F+F/F)F)7 'F)F-F*F.F)Q*pprint109F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "map(disp,map(inter,ST32,ST32[5]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7'K%'matrixG6#7$7'%\"~G<#\"\"%<#\"\"#F)F)7'F)<#\"\"\"<#\"\"&<#\"\"$F) Q*pprint1106\"KF%6#7$7'F)<\"<$F-F2F)F)7'F)<$F0F+F;F3F)Q*pprint111F6KF% 6#7$7'F)F*F;F)F)7'F)F/FF,F; F)Q*pprint113F6KF%6#7$7'F)F*F1F)F)7'F)F/F,F3F)Q*pprint114F6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "map(CC,ST32,ST32[5]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7'!\"\"\"\"!F%F%\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 105 "PCC(T1,T2) : Same as above, but displays the input and output \+ tableaux (bug: displays twice for bad int.)" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 39 "PCC := proc (A, B) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "local bool, sh, i, j, tetab, itab, te, k, s ss, beta, out;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sh := shape(A); k := nops(sh); tetab := inter(A,B); bool := true; itab := NULL;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for i to k while bool do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " te := NULL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " for j to sh[i] do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " sss := [ob(tetab[i][j])];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " if nops(sss) <> 1 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " out := 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " \+ print(disp(A),disp(B),` bad `);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " bool := false" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " \+ else te := te, ob(sss)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " fi " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " od; " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 21 " itab := itab, [te]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "if bool t hen " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " print(disp(A),disp(B),` \+ --> `,disp([itab]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " be ta := per([itab],B);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " out := s gn(beta)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "out" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ST32:=pck(ST5,[3,2]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "disp(inter(ST32[1],ST32[5])) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7'%\"~G<#\"\"%<#\" \"#F(F(7'F(<#\"\"\"<#\"\"&<#\"\"$F(Q*pprint1156\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "PCC(ST32[1],ST32[5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&K%'matrixG6#7$7'%\"~G\"\"#\"\"%F(F(7'F(\"\"\"\"\"$\"\"& F(Q*pprint1166\"KF$6#7$7'F(F*F.F(F(7'F(F,F)F-F(Q*pprint117F0%,~~~~-->~ ~~~GKF$6#7$7'F(F*F)F(F(7'F(F,F.F-F(Q*pprint118F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "PCC (ST32[2],ST32[5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%K%'matrixG6#7$7' %\"~G\"\"#\"\"&F(F(7'F(\"\"\"\"\"$\"\"%F(Q*pprint1196\"KF$6#7$7'F(F.F* F(F(7'F(F,F)F-F(Q*pprint120F0%)~~~bad~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%K%'matrixG6#7$7'%\"~G\"\"#\"\"&F(F(7'F(\"\"\"\"\"$\"\"%F(Q*pprin t1216\"KF$6#7$7'F(F.F*F(F(7'F(F,F)F-F(Q*pprint122F0%)~~~bad~~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 100 "CMAT(L1, L2) : For L1, L2 lists of tableaux, returns the C(L1,L2) matrix (C[i,j] = CC(L1[i],L2[j]);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "CMAT:=proc(L1,L2)\nlocal mat,i,j,ss,N,te,out;\nN: =nops(L1);\nout:=NULL;\n for i from 1 to N do\n te:=NULL;\n for j \+ from 1 to N do\n ss:=CC(L1[i],L2[j]);\n te:=te,ss;\n \+ od;\n out:=out,[te];\n od;\narray([out]);\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ST32:=pck(ST5,[3,2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "CMAT(ST32,ST32);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7'\"\"\"\"\"!F)F)!\"\"7'F)F(F)F)F)7'F)F )F(F)F)7'F)F)F)F(F)7'F)F)F)F)F(Q*pprint1236\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 120 "prep( la) : Creates global variables TABS and CID, to prepare to work with a rep of shape lambda (CID is returned) (more)" }}{PARA 4 "" 0 "" {TEXT -1 88 " TABS is a list of all standard tableaux of shape lambd a (in Young First Letter Order)" }}{PARA 4 "" 0 "" {TEXT -1 51 " CID is the inverse of the matrix CMAT(TABS,TABS)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "prep:=proc(la)\nlocal n,te,i;\nglobal TABS,CID; \nn:=size(la);\nte:=maddo([[[1]]]);\nfor i from 1 to n-2 do\n te:=mad do(te);\n od;\nTABS:=pck(te,la);\nCID:=inverse(CMAT(TABS,TABS));\nen d:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "prep([3,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7 %F)F)F(Q*pprint1246\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ma p(disp,TABS);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%K%'matrixG6#7$7'%\" ~G\"\"#F)F)F)7'F)\"\"\"\"\"$\"\"%F)Q*pprint1256\"KF%6#7$7'F)F-F)F)F)7' F)F,F*F.F)Q*pprint126F0KF%6#7$7'F)F.F)F)F)7'F)F,F*F-F)Q*pprint127F0" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 109 "NAT(sig) : For the \"prep\"ed partition lambda, retur ns A^lambda (sig) in Young's natural representation of S_n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "NAT:=proc(sig)\nlocal tmat,sigtabs ;\nsigtabs:=secact(sig,TABS);\ntmat:=CMAT(TABS,sigtabs);\nmultiply(CID ,tmat);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "prep([2,2,1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "NAT([1,3,2,5,4]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7'\"\"\"\"\"!F)F)F)7'F)F)F)F)F(7'F)F )F)F(F)7'F)F)F(F)F)7'F)F(F)F)F)Q*pprint1286\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 119 "gaAA( lamda) : Returns the entire (Young's natural) representation A^lambda \+ as a matrix of group algebra elements (more)" }}{PARA 4 "" 0 "" {TEXT -1 87 "Eg., to read off A^lamda(sig), look at the coefficient of x_sig in each matrix position" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 248 "gaAA:=proc(la)\nlocal i,j,dim,mat,rw,pers,sig,n;\noptions remember; \+ \nn:=size(la); \ndim:=fla(la);\nmat:=mkzmat(dim);\nprep(la);\npe rs:=permute([seq(i,i=1..n)]);\nfor sig in pers do\n mat:=matadd(mat, NAT(sig),1,xsig(sig));\n od;\nmatrix(mat);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "XX:=gaAA([2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXGK%'matrixG6#7$7$,*&%\"xG6%\"\"\"\"\"#\"\"$F.&F,6% F/F.F0!\"\"&F,6%F/F0F.F3&F,6%F0F/F.F.,*&F,6%F.F0F/F.F1F3F4F3&F,6%F0F.F /F.7$,*F9F.F4F.F;F3F6F3,*F+F.F1F.F;F3F6F3Q*pprint1296\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "NAT([3,1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7$\"\"!\"\"\"7$!\"\"F+Q*pprint1306\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "The Fouri er Transform for S_n" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 126 "mkgel(Pa irList) :Returns the corresponding group alg. element; Pairlist is a l ist of pairs, [a,b], of coeffs and perms in S_n " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 158 "mkgel:=proc(pairs)\nlocal i,n,f,coe,sig;\nf:= 0;\nn:=nops(pairs);\nfor i from 1 to n do\n coe:=pairs[i][1];\n sig: =pairs[i][2];\n f:=f+coe*xsig(sig);\n od;\nf;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "test:=[[a,[1,2,3,4]],[u,[2,1,3,4]]] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG7$7$%\"aG7&\"\"\"\"\"#\" \"$\"\"%7$%\"uG7&F*F)F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "F:=mkgel(test);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,&*&%\"aG \"\"\"&%\"xG6&F(\"\"#\"\"$\"\"%F(F(*&%\"uGF(&F*6&F,F(F-F.F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 111 "mkpairs(f, n) : Returns a list of pairs, [a,b], of coe ffs and permutations in S_n; f is a group algebra element" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "mkpairs:=proc(f,n)\nlocal pers,coe s,sigs,sig,coe,i,out,pars,la;\npers:=permute([seq(i,i=1..n)]);\nout:=N ULL;\nfor sig in pers do\n coe:=coeff(f,xsig(sig),1);\n if not coe=0 then out:=out,[coe,sig];fi;\n od;\n[out];\nend:\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "F:=mkgel([[a,[1,2,3,4]],[u,[ 2,1,3,4]]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,&*&%\"aG\"\"\"& %\"xG6&F(\"\"#\"\"$\"\"%F(F(*&%\"uGF(&F*6&F,F(F-F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "mkpairs(F,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$%\"aG7&\"\"\"\"\"#\"\"$\"\"%7$%\"uG7&F(F'F)F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 125 "fouriercoe(f,lambda) : Returns the Fourier coefficient for the group algebra element f, corresponding to the partition lambd a" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "fouriercoe:=proc(f,la) \nlocal pairs,sig,coe,i,mat,dim,AA,n;\nn:=size(la);\npairs:=mkpairs(f, n);\nprep(la);\ndim:=fla(la);\nmat:=mkzmat(dim);\nfor i from 1 to nops (pairs) do\n coe:=pairs[i][1];\n sig:=pairs[i][2];\n AA :=transpose(NAT(invsig(sig)));\n mat:=matadd(mat,AA,1,coe); \+ \n od; \nmatrix(mat);\nend:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "F:=mkgel([[a,[1,2,3,4]],[u,[2,1,3,4]]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,&*&%\"aG\"\"\"&%\"xG6&F(\"\"#\"\"$\" \"%F(F(*&%\"uGF(&F*6&F,F(F-F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fc31:=fouriercoe(F,[3,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fc31GK%'matrixG6#7%7%,&%\"aG\"\"\"%\"uG!\"\"\"\"!F/7 %,$F-F.,&F+F,F-F,F/7%F1F/F2Q*pprint1316\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 147 "fourier( f,n) : Returns a list of ordered pairs [a,b] where a is a partition of n and b is the matrix in the ath block of the fourier transform of f " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "fourier:=proc(f,n)\nloc al pars,la,out;\npars:=Par(n);\nout:=NULL;\nfor la in pars do\n out:= out,[la,fouriercoe(f,la)];\n od;\n[out];\nend:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "F:=mkgel([[a,[1,2,3,4]],[u,[2,1,3,4]]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,&*&%\"aG\"\"\"&%\"xG6&F(\"\"#\" \"$\"\"%F(F(*&%\"uGF(&F*6&F,F(F-F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "GG:=fourier(F,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#GGG7'7$7#\"\"%K%'matrixG6#7#7#,&%\"aG\"\"\"%\"uGF0Q*pprint1326\"7$7 $\"\"$F0KF*6#7%7%,&F/F0F1!\"\"\"\"!F=7%,$F1F " 0 "" {MPLTEXT 1 0 82 "for i from 1 to nops(GG) do print(i nverse(GG[i][2]));od; # We can invert matrices" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7#7#*&\"\"\"F),&%\"aGF)%\"uGF)!\"\"Q*pprint 1376\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%*&\"\"\"F),& %\"aGF)%\"uG!\"\"F-\"\"!F.7%*(F,F),&F+F)F,F)F-F*F-*&F)F)F1F-F.7%F0F.F2 Q*pprint1386\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7$*&\" \"\"F),&%\"aGF)%\"uG!\"\"F-\"\"!7$*(F,F),&F+F)F,F)F-F*F-*&F)F)F1F-Q*pp rint1396\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%*&\"\"\" F),&%\"aGF)%\"uG!\"\"F-\"\"!F.7%F.F(F.7%,$*(F,F),&F+F)F,F)F-F*F-F-F2*& F)F)F3F-Q*pprint1406\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7#7#*&\"\"\"F),&%\"aGF)%\"uG!\"\"F-Q*pprint1416\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 114 "invfourier(fourierlist) : Returns the group algebra element co rresponding to the matrix represented by fourierlist" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "invfourier:=proc(coes)\nlocal out,la,AA, BB,i,HLA;\nout:=0;\nfor i from 1 to nops(coes) do\n la:=coes[i][1];\n BB:=coes[i][2];\n AA:=gaAA(la);\n HLA:=hla(la);\n out:=out+dotpro d(AA,BB)/HLA;\n od;\nout;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "GG:=xsig([2,1,3])+a*x[3,2,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GGG,&&%\"xG6%\"\"#\"\"\"\"\"$F**&%\"aGF*&F'6%F+F)F*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fGG:=fourier(GG,3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fGGG7%7$7#\"\"$K%'matrixG6#7#7#,&\" \"\"F/%\"aGF/Q*pprint1426\"7$7$\"\"#F/KF*6#7$7$,&F/!\"\"F0F/,$F0F;7$F; ,&F/F/F0F;Q*pprint143F27$7%F/F/F/KF*6#7#7#,&F/F;F0F;Q*pprint144F2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "expand(invfourier(fGG));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"xG6%\"\"#\"\"\"\"\"$F(*&%\"aGF(& F%6%F)F'F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 89 "invelement(f,n) : Returns a fouri erlist; the inverse of the fourier transform of f in S_n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "invelement:=proc(f,n)\nlocal FT,i, out,IFT,la,coe;\nFT:=fourier(f,n);\nout:=NULL;\nfor i from 1 to nops(F T) do\n la:=FT[i][1];\n coe:=FT[i][2];\n out:=out,[la,inverse(co e)];\n od;\nIFT:=[out];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "G:=mkgel([[a,[1,2,3,4]],[b,[2,1,3,4]],[c,[1,2,4,3]]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG,(*&%\"aG\"\"\"&%\"xG6&F(\"\"#\" \"$\"\"%F(F(*&%\"bGF(&F*6&F,F(F-F.F(F(*&%\"cGF(&F*6&F(F,F.F-F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "H:=invelement(G,4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG7'7$7#\"\"%K%'matrixG6#7#7#*&\" \"\"F/,(%\"aGF/%\"cGF/%\"bGF/!\"\"Q*pprint1456\"7$7$\"\"$F/KF*6#7%7%*& F/F/,(F1F/F2F/F3F4F4\"\"!F@7%*(F3F/F0F4F?F4*&,&F1F/F3F/F/,**$)F1\"\"#F /F/*(FHF/F1F/F3F/F/*$)F3FHF/F/*$)F2FHF/F4F4,$*&F2F/FEF4F47%FBFNFCQ*ppr int146F67$7$FHFHKF*6#7$7$*&F/F/,(F1F/F2F4F3F4F4F@7$*(,&F3F/F2F/F/FYF4F 0F4F.Q*pprint147F67$7%FHF/F/KF*6#7%7%*&,&F1F/F3F4F/,*FFF/*(FHF/F1F/F3F /F4FJF/FLF4F4,$*&F2F/F`oF4F4F@7%FboF^oF@7%,$*&F3F/,*FFF/*(FHF/F1F/F2F/ F4FLF/FJF4F4F4Fgo*&F/F/,(F1F/F2F4F3F/F4Q*pprint148F67$7&F/F/F/F/KF*6#7 #7#FXQ*pprint149F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "facto r(invfourier(H));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,6*&)%\"aG\"\"$ \"\"\"&%\"xG6&F)\"\"#F(\"\"%F)F)*(%\"bGF)&F+6&F-F)F(F.F))F'F-F)!\"\"*( %\"cGF)&F+6&F)F-F.F(F)F3F)F4*,F-F)&F+6&F-F)F.F(F)F6F)F'F)F0F)F)*(F'F)F *F))F0F-F)F4*(F'F)F*F))F6F-F)F4*&)F0F(F)F1F)F)*(F0F)F1F)F?F)F4*&)F6F(F )F7F)F)*(F6F)F7F)F=F)F4F),(F'F)F6F4F0F)F4,(F'F)F6F4F0F4F4,(F'F)F6F)F0F )F4,(F'F)F6F)F0F4F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 35 "Group Det For the Dihedral Group D4" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "DG : Returns the Group D4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "DG:=[array([[1,0],[0,1]]),\narray([[1,0],[0,-1] ]),\narray([[0,1],[1,0]]),\narray([[0,1],[-1,0]]),\narray([[0,-1],[-1, 0]]),\narray([[0,-1],[1,0]]),\narray([[-1,0],[0,1]]),\narray([[-1,0],[ 0,-1]])]\n; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DGG7*K%'matrixG6#7 $7$\"\"\"\"\"!7$F,F+Q*pprint1506\"KF'6#7$F*7$F,!\"\"Q*pprint151F/KF'6# 7$F-F*Q*pprint152F/KF'6#7$F-7$F4F,Q*pprint153F/KF'6#7$F3F=Q*pprint154F /KF'6#7$F3F*Q*pprint155F/KF'6#7$F=F-Q*pprint156F/KF'6#7$F=F3Q*pprint15 7F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "same(A,B) : Tests for equality of two elements \+ of D4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "same:=proc(A,B)\nl ocal out;\nif A[1,1]=B[1,1] and\n A[1,1]=B[1,1] and\n A[1,2]=B[1,2] an d\n A[2,1]=B[2,1] and\n A[2,2]=B[2,2] \nthen out:=true else out:=false \nfi;\nout;\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "locateD(A) : Locates an element \+ of D4 in the canonical list" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "locateD:=proc(A)\n local out,i;\n for i from 1 to nops(DG) do\n if same(A,DG[i]) then out:=i; fi;\n od;\n out;\n end: " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "locateD(DG[4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 81 "GDETD( ) : Returns the matrix associated to computing the group determinant of D4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "GDETD:=proc(M)\n local out,i,alph a,beta,ibeta,rw,inde,gamma;\n out:=NULL;\n for alpha in DG do\n r w:=NULL;\n for beta in DG do\n ibeta:=inverse(beta);\n gamm a:=multiply(alpha,ibeta);\n inde:=locateD(gamma);\n rw:=rw,cat (x,inde);\n od;\n out:=out,[rw];\n od;\n array([out]);\n e nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "unassign('x1','x2',' x3','x4','x5','x6','x7','x8');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eg := GDETD();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#egGK%'mat rixG6#7*7*%#x1G%#x2G%#x3G%#x6G%#x5G%#x4G%#x7G%#x8G7*F+F*F/F.F-F,F1F07* F,F-F*F+F1F0F/F.7*F/F.F+F*F0F1F,F-7*F.F/F1F0F*F+F-F,7*F-F,F0F1F+F*F.F/ 7*F0F1F-F,F/F.F*F+7*F1F0F.F/F,F-F+F*Q*pprint1586\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "factor(det(eg));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,2%#x7G!\"\"%#x3G\"\"\"%#x6GF(%#x2GF&%#x1GF&%#x4GF(%# x5GF(%#x8GF&F(,2F%F(F'F(F)F&F*F(F+F&F,F&F-F(F.F&F(,2F%F(F'F(F)F(F*F(F+ F(F,F(F-F(F.F(F(,2F%F&F'F(F)F&F*F&F+F(F,F&F-F(F.F(F(),:*$)F.\"\"#F(F&* $)F*F6F(F(*$)F)F6F(F&*(F6F(F.F(F+F(F(*$)F+F6F(F&*(F6F(F)F(F,F(F(*$)F,F 6F(F&*(F6F(F-F(F'F(F&*$)F-F6F(F(*(F6F(F%F(F*F(F&*$)F%F6F(F(*$)F'F6F(F( F6F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "Group Det For the Alternating group A4" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "mkAn(n) : Returns the group An " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "mkAn:=proc(n)\nlocal sig,pers,out;\npers:=permute (n);\nout:=NULL;\nfor sig in pers do\n if sgn(sig)=1 then out:=out,si g; fi;\n od;\n[out];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "A4:=mkAn(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A4G7.7&\"\"\"\" \"#\"\"$\"\"%7&F'F)F*F(7&F'F*F(F)7&F(F'F*F)7&F(F)F'F*7&F(F*F)F'7&F)F'F (F*7&F)F(F*F'7&F)F*F'F(7&F*F'F)F(7&F*F(F'F)7&F*F)F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "locinG(A4[10],A4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "AnMat(n) : Returns the matr ix associated with the group det for An" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "AnMat:=proc(n)\nlocal pers,out,i,alpha,beta,ibeta,rw ,place,gamma;\npers:=mkAn(n);\nout:=NULL;\nfor alpha in pers do\n rw: =NULL;\n for beta in pers do\n ibeta:=invsig(beta);\n gamma:=mulp er(alpha,ibeta);\n place:=nlocit(gamma);\n rw:=rw,cat(x,place);\n \+ od;\n out:=out,[rw];\n od;\narray([out]);\nend:\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eg := AnMat(3);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#egGK%'matrixG6#7%7%%#x1G%#x5G%#x4G7%F,F*F+7%F+F,F* Q*pprint1596\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "factor(de t(eg));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(%#x4G\"\"\"%#x1GF&%#x5GF&F&,.*$)F%\"\"#F&F&*&F%F&F' F&!\"\"*&F(F&F%F&F.*&F(F&F'F&F.*$)F'F,F&F&*$)F(F,F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "toobig := AnMat(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'toobigGK%'matrixG6#7.7.%#x1G%#x5G%#x4G%#x8G%$x1 3G%$x20G%#x9G%$x21G%$x17G%$x12G%$x16G%$x24G7.F,F*F+F.F/F-F2F0F1F4F5F37 .F+F,F*F/F-F.F1F2F0F5F3F47.F-F0F3F*F1F4F+F.F5F,F/F27.F0F3F-F4F*F1F.F5F +F2F,F/7.F3F-F0F1F4F*F5F+F.F/F2F,7.F.F2F4F,F0F5F*F/F3F+F-F17.F4F.F2F0F 5F,F3F*F/F-F1F+7.F2F4F.F5F,F0F/F3F*F1F+F-7.F/F1F5F+F2F3F,F-F4F*F.F07.F 1F5F/F3F+F2F-F4F,F0F*F.7.F5F/F1F2F3F+F4F,F-F.F0F*Q*pprint1606\"" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 17 "Some Polya theory" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 131 "bpairtono(i,j,n) : (Assumes 1<=i " 0 "" {MPLTEXT 1 0 56 "bpairtono:=proc(i,j,n)\n1+i+ j+n*(i-1)-(i+1)*(i+2)/2;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "bpairtono(1,5,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 77 "pairtono(i,j,n) : Returns bpairtono(i,j,n) if i " 0 "" {MPLTEXT 1 0 112 "p airtono:=proc(i,j,n)\nlocal out;\nif i \+ " 0 "" {MPLTEXT 1 0 16 "pairtono(3,2,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 94 "mkpair(n) : Returns a list of all order ed pairs [i,j] such that 1<=i " 0 "" {MPLTEXT 1 0 189 "mkpair:=proc(n)\nlocal i,j,out,pai r;\noptions remember;\npair:=[seq(0,i=1..n*(n-1)/2)];\nfor i from 1 to n-1 \n do\n for j from i+1 to n do pair[bpairtono(i,j,n)]:=[i,j]; \n od;od;\npair;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "mkpair(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#717$\"\"\"\"\"#7$F% \"\"$7$F%\"\"%7$F%\"\"&7$F%\"\"'7$F&F(7$F&F*7$F&F,7$F&F.7$F(F*7$F(F,7$ F(F.7$F*F,7$F*F.7$F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "PRTNO([i,j],n) : Returns pair tono(i,j,n)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "PRTNO:=proc(p r,n)\npairtono(pr[1],pr[2],n);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "P6:=mkpair(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# P6G717$\"\"\"\"\"#7$F'\"\"$7$F'\"\"%7$F'\"\"&7$F'\"\"'7$F(F*7$F(F,7$F( F.7$F(F07$F*F,7$F*F.7$F*F07$F,F.7$F,F07$F.F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "map(PRTNO,P6,6); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#71\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"# 6\"#7\"#8\"#9\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 109 "matsig(sig) : Returns the matrix corresponding to the action of sig on the set of pairs with (i,j), 1< =i " 0 "" {MPLTEXT 1 0 359 "matsig:=proc(sig )\nlocal i,j,out,row,pat,N,pr,pairs,imi,n;\noptions remember;\nn:=nops (sig);\nN:=n*(n-1)/2;\npairs:=mkpair(n);\nout:=NULL;\nfor i from 1 to \+ N do\n pr:=pairs[i];\n row:=NULL;\n imi:=pairtono(sig[pr[1]],sig[pr [2]],n);\n for j from 1 to N do\n if j=imi then row:=row,1 el se row:=row,0; fi;\n od;\n out:=out,[row];\n od;\narray([ou t]);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "matsig([2,1,3 ,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7(\"\"\"\"\"!F) F)F)F)7(F)F)F)F(F)F)7(F)F)F)F)F(F)7(F)F(F)F)F)F)7(F)F)F(F)F)F)7(F)F)F) F)F)F(Q*pprint1616\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 114 "persig(sig) : Returns the perm utation corresponding to the action of sig on the set of pairs with (i ,j), 1<=i " 0 "" {MPLTEXT 1 0 242 "persig:=p roc(sig)\nlocal i,j,out,pat,N,pr,pairs,imi,n;\noptions remember;\nn:=n ops(sig);\nN:=n*(n-1)/2;\npairs:=mkpair(n);\nout:=NULL;\nfor i from 1 \+ to N do\n pr:=pairs[i];\n imi:=pairtono(sig[pr[1]],sig[pr[2]],n);\n \+ out:=out,imi;\n od;\n[out];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "persig([2,1,3,4]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"\"\"%\"\"&\"\"#\"\"$\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 78 "PAIR(i,n) : Returns the ith pair in the lex list of pairs [i,j] with 1<=i " 0 "" {MPLTEXT 1 0 61 "PAIR:=proc(i,n)\nlocal pair s;\npairs:=mkpair(n);\npairs[i];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "mkpair(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,7$\" \"\"\"\"#7$F%\"\"$7$F%\"\"%7$F%\"\"&7$F&F(7$F&F*7$F&F,7$F(F*7$F(F,7$F* F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PAIR(6,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ps:=persig([2,4,3,1,5]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#psG7,\"\"'\"\"&\"\"\"\"\"(\"\")\"\"$\"#5\"\"#\"\"*\" \"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "PermutedPairList:=ma p(PAIR,ps,5); # Show the pairs in permuted order after acting on by [2 ,4,3,1,5]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1PermutedPairListG7,7$ \"\"#\"\"%7$F'\"\"$7$\"\"\"F'7$F'\"\"&7$F*F(7$F,F(7$F(F.7$F,F*7$F*F.7$ F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "map(PRTNO,PermutedPairList,5); # Verify tha t persig did the right thing" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\" '\"\"&\"\"\"\"\"(\"\")\"\"$\"#5\"\"#\"\"*\"\"%" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 123 "cycindex(n) : Returns the \+ cycle index polynomial corresponding to the action of S_n on ordered p airs [i,j] with 1 " 0 "" {MPLTEXT 1 0 240 "cyclindex:=proc(n)\nlocal pol,prs,pers,sig,i,bisig,cyc,pow;\nprs: =mkpair(n);\npers:=permute([seq(i,i=1..n)]);\npol:=0;\nfor sig in pers do \n bisig:=persig(sig);\n cyc:=cycstr(bisig);\n pow:=mkpowsym (cyc);\n pol:=pol+pow;\n od;\npol/n!;\nend:" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "cyc:=cyclindex(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cycG,**&\"#C!\"\"%#p1G\"\"'\"\"\"**\"\"$F +\"\")F(%#p2G\"\"#F)F0F+*&F-F(%#p3GF0F+*(\"\"%F(%#p4GF+F/F+F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tom(cyc);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,8&%\"mG6#\"\"'\"\"\"*&\"\"#F(&F%6$\"\"%F*F(F(*&\"\"$ F(&F%6$F/F/F(F(*&\"\"&F(&F%6&F/F(F(F(F(F(*&F-F(&F%6%F/F*F(F(F(*&F*F(&F %6%F-F(F(F(F(*&F'F(&F%6%F*F*F*F(F(&F%6$F3F(F(*&\"\"*F(&F%6&F*F*F(F(F(F (*&\"#:F(&F%6'F*F(F(F(F(F(F(*&\"#IF(&F%6(F(F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Invariant T heory procedures" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 72 "T(A,P) : Retu rns the polynomial generated by acting on P by the matrix A" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "T:=proc(A,P)\nlocal n,te,out,i,j,p at;\nn:=rowdim(A);\npat:=NULL;\nfor j from 1 to n do\n te:=0;\n for \+ i from 1 to n do\n te:=te+(cat(x,i))*A[i,j];\n od;\n pat:=pat,(cat(x,j))=te;\n od;\nsubs(\{pat\},P);\nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "AAA:=mkmatA(3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AAAGK%'matrixG6#7%7%%$a11G%$a12G%$a 13G7%%$a21G%$a22G%$a23G7%%$a31G%$a32G%$a33GQ*pprint1626\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "T(AAA,x1*x2*x3);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*(,(*&%#x1G\"\"\"%$a11GF'F'*&%#x2GF'%$a21GF'F'*&%#x3G F'%$a31GF'F'F',(*&F&F'%$a12GF'F'*&F*F'%$a22GF'F'*&F-F'%$a32GF'F'F',(*& F&F'%$a13GF'F'*&F*F'%$a23GF'F'*&F-F'%$a33GF'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Poly:=x1^3*x2^2*x3^5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PolyG*()%#x1G\"\"$\"\"\")%#x2G\"\"#F))%#x3G\"\"&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "T(AAA,Poly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(),(*&%#x1G\"\"\"%$a11GF(F(*&%#x2GF(%$a21GF(F( *&%#x3GF(%$a31GF(F(\"\"$F(),(*&F'F(%$a12GF(F(*&F+F(%$a22GF(F(*&F.F(%$a 32GF(F(\"\"#F(),(*&F'F(%$a13GF(F(*&F+F(%$a23GF(F(*&F.F(%$a33GF(F(\"\"& F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "charpoly(AAA,q);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,B*$)%\"qG\"\"$\"\"\"F(*&)F&\"\"#F(%$a 33GF(!\"\"*&%$a22GF(F*F(F-*(F&F(F/F(F,F(F(*(F&F(%$a23GF(%$a32GF(F-*&%$ a11GF(F*F(F-*(F5F(F&F(F,F(F(*(F5F(F/F(F&F(F(*(F5F(F/F(F,F(F-*(F5F(F2F( F3F(F(*(%$a21GF(%$a12GF(F&F(F-*(F;F(F " 0 "" {MPLTEXT 1 0 32 "H3:=expand(evalsf(h3,x1+x2+x3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H3G,6*$)%#x1G\"\"$\"\"\"F**$)%#x2GF)F*F**$)% #x3GF)F*F**&F(F*)F-\"\"#F*F**&F(F*)F0F3F*F**&F-F*)F(F3F*F**&F-F*F5F*F* *&F0F*F7F*F**&F0F*F2F*F**(F(F*F-F*F0F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "BBB:=mkpermat([3,2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BBBGK%'matrixG6#7%7%\"\"!F*\"\"\"7%F*F+F*7%F+F*F*Q*pprint1636 \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "H3-T(BBB,H3); # H3 i s invariant under symmetric functions" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 89 "REY(G,P) : Returns a polynomial correspon ding to the Reynolds operator for G applied to P" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 128 "REY:=proc(G,P)\nlocal out,i,N;\nN:=nops(G);\n out:=0;\nfor i from 1 to N do\n out:=out+T(G[i],P);\n od; \nexpand(out/N);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "S 4:=permute(4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "GS4:=map( mkpermat,S4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Poly:=x1+x 2^2+x3^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PolyG,(%#x1G\"\"\"*$)% #x2G\"\"#F'F'*$)%#x3G\"\"$F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "REY(GS4,Poly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,:*&\"\"%!\" \"%#x1G\"\"\"F(*&F%F&%#x2G\"\"#F(*&F%F&%#x3G\"\"$F(*&F%F&%#x4GF.F(*&F% F&F-F+F(*&F%F&F*F.F(*&F%F&F0F+F(*&F%F&F*F(F(*&F%F&F'F+F(*&F%F&F'F.F(*& F%F&F-F(F(*&F%F&F0F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 139 "chara(lambda,rho) : Returns \+ the character of A^(lambda) on the conj class corresponding to rho, us es sym. functs. (lam, rho must be sorted)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "chara:=proc(la,rho)\nlocal out,sch;\noptions remembe r;\nsch:=tos(mkpowsym(rho));\nout:=coeff(sch,s[op(la)]);\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 103 "char(lam bda,rho) : Returns character of A^(lambda) at rho conj class, lambda a nd rho need not be sorted" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "char:=(lambda,rho)->chara(par(lambda),par(rho)):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "char([3,1,1,1],[2,2,1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 65 "chilasig(lambda,sig) : Return s the character of A^(lambda) at sig" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "chilasig:=(lambda,sig)->char(lambda,cycstr(sig)):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "chilasig([2,2,1,1,1],[4,1,3 ,7,2,5,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 185 "hilb(G) : Returns the Hilbert series for the group G (where G is a list of matrices). This is done using Moliens Theorem: The hilbe rt series = (1/|G|) * Sum(A in G) [1 / det( I - q*A )]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "hilb:=proc(mats)\nlocal A,out,N;\n N:=nops(mats); \nout:=0;\nfor A in mats do\n out:=out+1/mycharpol(A); \n od;\nout/N;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "p erm3:=permute(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&perm3G7(7%\"\" \"\"\"#\"\"$7%F'F)F(7%F(F'F)7%F(F)F'7%F)F'F(7%F)F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "S3:=map(matsig,perm3); # matsig returns the matrix corresponding to the action on ordered pairs, NOT the perm utation matrix" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S3G7(K%'matrixG6# 7%7%\"\"\"\"\"!F,7%F,F+F,7%F,F,F+Q*pprint1646\"KF'6#7%F-F*F.Q*pprint16 5F0KF'6#7%F*F.F-Q*pprint166F0KF'6#7%F.F*F-Q*pprint167F0KF'6#7%F-F.F*Q* pprint168F0KF'6#7%F.F-F*Q*pprint169F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "FRS3:=h ilb(S3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%FRS3G,(*&\"\"\"F'*&\"\" 'F',*F'F'*&\"\"$F'%\"qGF'!\"\"*&F,F')F-\"\"#F'F'*$)F-F,F'F.F'F.F'*&F'F '*&F1F',*F2F'*$F0F'F.F-F.F'F'F'F.F'*&F'F'*&F,F',&F'F'F2F.F'F.F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "factor(FRS3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*(,&%\"qGF%F%F%F%),&F(F%F%!\"\"\"\"$ F%,(*$)F(\"\"#F%F%F(F%F%F%F%F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "S22:=conca(permute([1,2]),permute([3,4]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S22G7&7&\"\"\"\"\"#\"\"$\"\"%7&F'F(F*F)7& F(F'F)F*7&F(F'F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "matS2 2:=map(matsig,S22);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'matS22G7&K%' matrixG6#7(7(\"\"\"\"\"!F,F,F,F,7(F,F+F,F,F,F,7(F,F,F+F,F,F,7(F,F,F,F+ F,F,7(F,F,F,F,F+F,7(F,F,F,F,F,F+Q*pprint1706\"KF'6#7(F*F.F-F0F/F1Q*ppr int171F3KF'6#7(F*F/F0F-F.F1Q*pprint172F3KF'6#7(F*F0F/F.F-F1Q*pprint173 F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "FRS22:=hilb(matS22); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&FRS22G,&*&\"\"\"F'*&\"\"%F',0*$ )%\"qG\"\"'F'F'*&F.F')F-\"\"&F'!\"\"*&\"#:F')F-F)F'F'*&\"#?F')F-\"\"$F 'F2*&F4F')F-\"\"#F'F'*&F.F'F-F'F2F'F'F'F2F'*(F9F'F)F2,0F+F'*&F " 0 "" {MPLTEXT 1 0 14 "factor(FRS22);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,(*$)%\"qG\"\"#\"\"\"F)F'!\"\"F)F)F),&F'F)F)F)!\"#,&F'F)F)F*! \"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "S32:=conca(permute([ 1,2,3]),permute([4,5]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S32G7.7 '\"\"\"\"\"#\"\"$\"\"%\"\"&7'F'F(F)F+F*7'F'F)F(F*F+7'F'F)F(F+F*7'F(F'F )F*F+7'F(F'F)F+F*7'F(F)F'F*F+7'F(F)F'F+F*7'F)F'F(F*F+7'F)F'F(F+F*7'F)F (F'F*F+7'F)F(F'F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "matS 32:=map(matsig,S32):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "FRS 32:=hilb(matS32);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&FRS32G,,*&\"\" \"F'*&\"#7F',8*$)%\"qG\"#5F'F'*&F.F')F-\"\"*F'!\"\"*&\"#XF')F-\"\")F'F '*&\"$?\"F')F-\"\"(F'F2*&\"$5#F')F-\"\"'F'F'*&\"$_#F')F-\"\"&F'F2*&FF',2F+F '*$F0F'F2*&FHF'F9F'F2*&FHF'F=F'F'*&FHF'FDF'F'*&FHF'FGF'F2F-F2F'F'F'F2F '*&F'F'*&F>F',2F'F'F-F2*$FGF'F2*$FDF'F'*$F=F'F2*$F9F'F'FboF'F+F2F'F2F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(FRS32);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*,,8*&\"\"#\"\"\")%\"qG\"#5F'F'*&\"\"$ F')F)\"\"*F'F'*&\"\"&F')F)\"\")F'F'*&\"\"%F')F)\"\"(F'F'*&F*F')F)\"\"' F'F'*&F&F')F)F0F'F'*&F4F')F)F4F'F'*&F4F')F)F,F'F'*&F&F')F)F&F'F'F)!\" \"F'F'F',&F)F'F'F'!\"%,(*$FAF'F'F)FBF'F'FB,(FFF'F)F'F'F'!\"$,&F)F'F'FB !#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 184 "idempREY(P,lambda) : Returns [(1/h_lambda) * S um_(sig in Sn) (Chi^lambda (sig) * sig P)] (The action of an irreducib le idempotent of Sn on P) (Note: I think h_lambda should be n!, -JB)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "idempREY:=proc(P,la)\nloc al out,i,n,pers,sig,A,ch,rho,TAP;\nn:=size(la);\npers:=permute([seq(i, i=1..n)]);\nout:=0;\nfor sig in pers do\n A:=mkpermat(sig);\n rho:=c ycstr(sig);\n ch:=char(la,rho);\n TAP:=T(A,P);\n out:=out+ch*TAP/hl a(la);\n od;\nout;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P:=x1*x2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG*&%#x1G\"\"\"%#x2G F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "P1:=factor(idempREY(P ,[3,2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G,6*(\"\"#!\"\"%#x1G \"\"\"%#x2GF*F**(\"\"'F(F)F*%#x3GF*F(*(F-F(F)F*%#x4GF*F(*(F-F(F)F*%#x5 GF*F(*(F-F(F.F*F+F*F(*(F-F(F+F*F0F*F(*(F-F(F+F*F2F*F(*(F-F(F.F*F0F*F** (F-F(F.F*F2F*F**(F-F(F0F*F2F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P2:=idempREY(P1,[3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# P2G,6*(\"\"#!\"\"%#x1G\"\"\"%#x2GF*F**(\"\"'F(F)F*%#x3GF*F(*(F-F(F)F*% #x4GF*F(*(F-F(F)F*%#x5GF*F(*(F-F(F.F*F+F*F(*(F-F(F+F*F0F*F(*(F-F(F+F*F 2F*F(*(F-F(F.F*F0F*F**(F-F(F.F*F2F*F**(F-F(F0F*F2F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(P1-P2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 271 "idemhilb(lambda) : Returns [(1 /h_lambda) * Sum(sig in Sn) [Chi^lambda (sig) / det(I-q*A)] (The hilbe rt series of the image of Q[Xn] under the action of the irreducible id empotent of Sn corresponding to the character of A^lambda) (Note: I th ink h_lambda should be n!, -JB)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 248 "idemhilb:=proc(la)\nlocal A,out,n,pers,sig,rho,ch;\nn:=size(l a); \npers:=permute([seq(i,i=1..n)]);\nout:=0;\nfor sig in pers do\n \+ A:=mkpermat(sig);\n rho:=cycstr(sig);\n ch:=char(la,rho);\n out:=ou t+ch/mycharpol(A)/hla(la); \n od;\nout;\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "boo:=factor(idemhilb([3,2])) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$booG,$*.\"\"&\"\"\"%\"qG\"\"#, &F)F(F(F(!\"#,(*$)F)F*F(F(F)F(F(F(!\"\",&F)F(F(F0!\"&,&F.F(F(F(F0F0" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "baa:=factor(evalsf(s[3,2], 1/(1-q)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$baaG,$*,%\"qG\"\"#,&F '\"\"\"F*F*!\"#,(*$)F'F(F*F*F'F*F*F*!\"\",&F'F*F*F/!\"&,&F-F*F*F*F/F/ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "series(boo,q,20);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+I%\"qG\"\"&\"\"#\"#5\"\"$\"#?\"\"%\"# NF%\"#g\"\"'\"#!*\"\"(\"$N\"\"\")\"$!>\"\"*\"$l#F'\"$b$\"#6\"$q%\"#7\" $0'\"#8\"$v(\"#9\"$q*\"#:\"%07\"#;\"%v9\"#<\"%&z\"\"#=\"%b@\"#>-%\"OG6 #\"\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 119 "mkbasis(GB,al,d) : makes a basis from a \+ gbasis, an alphabet and a degree (THIS DOESN'T WORK; missing procedure testmon)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 424 "mkbasis:=proc( GB,al,d)\nlocal k,out,lmons,mons,sal,moes,i,mon,plmons;\nsal:=convert( al,`+`);\nout:=NULL;\nplmons:=map(leadmon,GB,al);\nlmons:=NULL;\nfor i from 1 to nops(plmons) do\n lmons:=lmons,plmons[i][2];\n od;\nl mons:=sort([lmons]);\nfor k from 1 to d do\n coeffs(expand(sal^k),al, 'moes');\n mons:=sort([moes]);\n for mon in mons do \n if not t estmon(mon,lmons) then \n out:=out,mon; fi;\n od;\n od;\n[ou t];\nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 100 "get mons(vars,deg) outputs a list of all monomials in a chosen number of v ariables of a chosen degree" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 335 "getmons := proc(vars, deg) local M, C, CC, P, i, Sn, SS;\nP := Pa r(deg, vars); Sn := permute(vars); M := map(sigact,Sn,map(mkmon2,P) ); C := map(convert,M,set); CC := convert(C,array); for i from 2 to nops (C) do; CC[1] := CC[1] union CC[i]; od;\nSS := sort(convert([op(CC[1]) ],`+`),[seq(cat(x,i),i=1..vars)],plex); convert(SS,list);end: " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "# getmons(Garsia's):=proc(n ,d)\n# local out, i, al, sal, mose , cose;\n# al:= [ seq(x||i, i=1 .. \+ n)];\n# sal := convert([ seq(x||i,i=1..n)],`+`);\n# cose:= coeffs(expa nd(sal^d),al, 'mose');\n# out:=[mose];\n# end:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Mono := getmons(3,7); nops(Mono);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%MonoG7F*$)%#x1G\"\"(\"\"\"*&)F(\"\"'F*%#x 2GF**&F,F*%#x3GF**&)F(\"\"&F*)F.\"\"#F**(F2F*F.F*F0F**&F2F*)F0F5F**&)F (\"\"%F*)F.\"\"$F**(F:F*F4F*F0F**(F:F*F.F*F8F**&F:F*)F0F=F**&)F(F=F*)F .F;F**(FCF*F " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 144 "Allinvar(G, deg, m) : creates a regular \+ sequence homogoeneous invariants of group G of degree = deg \n(Assumes Group G given as m x m matrices) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 702 "Allinvar := proc(G, deg,m) local Mons, i,j,k, S, SS , n, E, L, hold, evalu, test; \nS := \{\}; Mons := getmons(m,deg); hol d := 0;\nfor j from 1 to nops(Mons) do;\nif REY(G,Mons[j] ) <> 0 then \+ if hold = 0 then S := S union \{REY(G,Mons[j] )\}; hold := j; break; f i; fi; od;\nif S <> \{\} then \nfor j from hold+1 to nops(Mons) do; SS := [seq(0,i=1..nops(S))]; for i from 1 to nops(S) do; SS[i] := S[i ]-cat(y,i); od; E := testAI( REY(G,Mons[j] ), SS,m); \nif E = 0 then L := y1; else L:= leadmon(E,[seq(cat(x,i),i=1..m),seq(cat(y,i),i=1..nop s(SS))],plex); fi; \nevalu := eval(L,[seq(cat(x,i)=0,i=1..m)]); \ntest := type(evalu[2],`numeric`); if test = 'true' then S := S union \{REY (G,Mons[j])\} fi; od; fi; S; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "D6 := mkdihedr(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#D6G7.K%'matrixG6#7$7$\"\"\"\"\"!7$F,F+Q*pprint1746\"KF'6#7$7$#F+\" \"#,$*&F5!\"\"\"\"$F4F87$,$*&F5F8F9F4F+F4Q*pprint175F/KF'6#7$7$#F8F5F6 7$F;FBQ*pprint176F/KF'6#7$7$F8F,7$F,F8Q*pprint177F/KF'6#7$7$FBF;7$F6FB Q*pprint178F/KF'6#7$7$F4F;7$F6F4Q*pprint179F/KF'6#7$F*FIQ*pprint180F/K F'6#7$F3FOQ*pprint181F/KF'6#7$FAFUQ*pprint182F/KF'6#7$FHF-Q*pprint183F /KF'6#7$FNF:Q*pprint184F/KF'6#7$FTFCQ*pprint185F/" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "Allinvar(D6,6,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,**&#\"\"*\"#K\"\"\"*$)%#x2G\"\"'F)F)F)*&#\"#6F(F)*$) %#x1GF-F)F)F)*&#\"#:F(F)*&)F3\"\"%F))F,\"\"#F)F)F)*&#\"#XF(F)*&)F3F;F) )F,F9F)F)F),**&#\"\"$F(F)F*F)F)*&#F)F(F)F1F)F)*&#\"#@F(F)F7F)F)*&#F'F( F)F?F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 314 "Allinvar2(G, deg) : creates a set of homogoeneous invariants o f group G of degree = deg (Assumes Group G given as 2x2 matrices)\nUse s rudimentary Jacobian test to eliminate repeats and obvious alg depen decies but does not nec. give minimal set. But will always give a spa nning set\nFASTER than Allinvar for deg > 10" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "Allinvar2 := proc(G, deg) local i,j,k,l, S, J,n; \nS := \{\}; for j from 0 to deg do;\nn := nops(S); J := []; for k fr om 1 to n do; J := [op(J), factor(det(jacobian([S[k],REY(G,mkmon2([j ,deg-j]))],[x1,x2])))]; od;\nif convert(J, `*`) <> 0 then S := S union \{REY(G,mkmon2([j,deg-j]))\} fi; \n od; S; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Allinvar2(D6,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&,**&#\"#6\"#K\"\"\"*$)%#x2G\"\"'F)F)F)*&#\"\"*F(F)*$) %#x1GF-F)F)F)*&#\"#XF(F)*&)F3\"\"%F))F,\"\"#F)F)F)*&#\"#:F(F)*&)F3F;F) )F,F9F)F)F),**&F/F)F*F)F)*&F&F)F1F)F)*&F=F)F7F)F)*&F5F)F?F)F),**&#F)F( F)F*F)F)*&#\"\"$F(F)F1F)F)*&#F0F(F)F7F)!\"\"*&#\"#@F(F)F?F)F),**&FKF)F *F)F)*&FIF)F1F)F)*&FQF)F7F)F)*&#F0F(F)F?F)FO" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 313 "Allin var3(G, deg) : creates a set of homogoeneous invariants of group G of \+ degree = deg (Assumes Group G given as 3x3 matrices)\nUses rudimentary Jacobian test to eliminate repeats and obvious alg dependecies but do es not nec. give minimal set. But will always give a spanning set\nFA STER than Allinvar for deg > 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 559 "Allinvar3 := proc(G, deg) local i,j,k,l, S, J,n; \nS := \{\}; \+ for i from 0 to deg do; for j from 0 to deg-i do;\nn := nops(S); J := \+ []; for k from 1 to n do; J := [op(J), factor(det(jacobian([S[k],REY (G,mkmon2([j,i,deg-j-i]))],[x1,x2])))]; od;\nfor k from 1 to n do; J \+ := [op(J), factor(det(jacobian([S[k],REY(G,mkmon2([j,i,deg-j-i]))],[x 1,x3])))]; od;\nfor k from 1 to n do; J := [op(J), factor(det(jacobi an([S[k],REY(G,mkmon2([j,i,deg-j-i]))],[x2,x3])))]; od;\nif convert(J, `*`) <> 0 then S := S union \{REY(G,mkmon2([j,i,deg-j-i]))\} fi; \n od; od; S; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "A4;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7.7&\"\"\"\"\"#\"\"$\"\"%7&F%F'F(F&7&F %F(F&F'7&F&F%F(F'7&F&F'F%F(7&F&F(F'F%7&F'F%F&F(7&F'F&F(F%7&F'F(F%F&7&F (F%F'F&7&F(F&F%F'7&F(F'F&F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "prep([2,1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7% \"\"\"\"\"!F)7%F)F(F)7%F)F)F(Q*pprint1866\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "AA4 := map(NAT,A4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AA4G7.K%'matrixG6#7%7%\"\"\"\"\"!F,7%F,F+F,7%F,F,F+Q*pprint18 76\"KF'6#7%7%F,!\"\"F,7%F,F,F5F*Q*pprint188F0KF'6#7%F.7%F5F,F,F4Q*ppri nt189F0KF'6#7%7%F,F5F57%F5F,F+F6Q*pprint190F0KF'6#7%7%F+F+F,F@F-Q*ppri nt191F0KF'6#7%FAFFF;Q*pprint192F0KF'6#7%7%F+F,F5F.F@Q*pprint193F0KF'6# 7%7%F5F5F,F*FAQ*pprint194F0KF'6#7%7%F,F+F+F4FFQ*pprint195F0KF'6#7%F6FY FOQ*pprint196F0KF'6#7%F-FTFYQ*pprint197F0KF'6#7%F;FOFTQ*pprint198F0" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Allinvar3(AA4,4);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<%,@*&#\"\"\"\"\"#F'*$)%#x3G\"\"%F'F'F '*&F&F'*$)%#x2GF,F'F'F'*&F&F'*$)%#x1GF,F'F'F'*&#\"\"$F(F'*&)F0F(F')F4F (F'F'F'*&)F0F7F'F4F'!\"\"*&)F4F7F'F0F'F=*&F6F'*&)F+F(F'F:F'F'F'**F7F'F :F'F0F'F+F'F=**F7F'F4F'F+F'F9F'F'**F7F'F4F'F0F'FBF'F=*&F+F'F?F'F'*&F4F ')F+F7F'F'*&F+F'FF'F=*&FOF'FFF'F'*&#F'F7F 'FPF'F=*&#F'F7F'FUF'F=*&FOF'FGF'F'*&#F'F7F'FJF'F=*&FhnF'F)F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "# Both this function and Al linvar2 work by applying the REY operator to all appropriate monomials \n# and add the answer to the set of invariants if and only if th e Jacobian of it with previous ones is nontrivial\n# Allinvars is more sophisticated and uses testAI to make sure don't add any element that was already in ideal" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 154 "Everyinvar(G,deg,m) : creates spanning set of invariants very quickly by applying Reynolds operator to all monomials. Very far from minimal set but FAST." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "Everyinvar := proc(G, deg,m) local Mons, i,j,k, S, SS, n, E, L, hold, evalu, test; \nMons := getmons(m, deg); L := [seq(0,i=1..nops(Mons))]; for j from 1 to nops(Mons) do; L[ j] := REY(G,Mons[j] ); od; L; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Everyinvar(D6,6,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7),**&#\"\"*\"#K\"\"\"*$)%#x2G\"\"'F)F)F)*&#\"#6F(F)*$)%#x1GF-F)F)F) *&#\"#:F(F)*&)F3\"\"%F))F,\"\"#F)F)F)*&#\"#XF(F)*&)F3F;F))F,F9F)F)F)\" \"!,**&#\"\"$F(F)F*F)F)*&#F)F(F)F1F)F)*&#\"#@F(F)F7F)F)*&#F'F(F)F?F)! \"\"FB,**&FHF)F*F)F)*&FEF)F1F)F)*&#F'F(F)F7F)FN*&FJF)F?F)F)FB,**&F/F)F *F)F)*&F&F)F1F)F)*&F=F)F7F)F)*&F5F)F?F)F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "Everyinvar(AA4,6,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 141 "SepPoly(et a,ideal,vars) : Returns the polynomial satisfied by eta and the elemen ts of the ideal assuming the right number (vars) of variables" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 492 "SepPoly := proc(eta,ideal,v ars) local i,k, BB, GB, E, test, P;\nP := \{\}; BB := ideal; for i fr om 1 to nops(ideal) do; BB[i] := ideal[i]-cat(y,i); od;\nGB := gbasis( [op(BB), eta - y ], [seq(cat(x,i),i=1..vars),seq(cat(y,k),k=1..nops(id eal)), y]);\nE := eval(map(leadmon,GB, [seq(cat(x,i),i=1..vars),seq(ca t(y,k),k=1..nops(ideal)), y]),[seq(cat(x,i)=0,i=1..vars)]); \ntest := \+ map(type,E,`numeric`); for k from 1 to nops(test) do; if test[k] = 'fa lse' then P := P union \{GB[k/2]\}; fi; od; P; end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 386 "ETA := 7/12*x2^2*x3^3*x1-1/4*x2^3*x3^2*x1-1/4*x2*x3^ 3*x1^2-7/12*x2^3*x1^2*x3-7/12*x2*x3^2*x1^3+1/4*x2^2*x1^3*x3-1/12*x2*x3 ^5+1/3*x1^4*x3^2+1/12*x1^5*x3-1/12*x1*x3^5+1/3*x2^4*x1^2-1/12*x2^2*x1^ 4+1/12*x1^5*x2+1/12*x3*x2^5-1/12*x3^2*x2^4-1/12*x1*x2^5-1/4*x3^3*x2^3- 1/12*x3^4*x1^2-1/12*x3^4*x1*x2+1/12*x1*x3*x2^4+1/4*x1^3*x3^3+3/4*x1^2* x2^2*x3^2-1/12*x1^4*x2*x3-1/4*x1^3*x2^3+1/3*x3^4*x2^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "I2 := 1/2*x1^2+1/2*x2^2+1/2*x3^2-1/ 2*x1*x2+1/2*x1*x3-1/2*x2*x3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "I3 := -3/4*x2*x1^2+3/4*x3*x1^2+3/4*x1*x2^2-3/2*x1*x2*x3+3/4*x1*x 3^2+3/4*x3*x2^2-3/4*x2*x3^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "I4 := 1/2*x2^4+1/2*x1^4+1/2*x3^4+3/2*x2^2*x3^2+3/2*x1^2*x3^2+3/ 2*x1^2*x2^2-x2^3*x1+x3^3*x1-x2^3*x3+3*x3*x2^2*x1-3*x2*x3^2*x1-3*x1^2*x 2*x3-x3^3*x2+x1^3*x3-x1^3*x2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# SepPoly(ETA,[I2,I3,I4],3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "\{36*y1^6-160*y1^3*y2^2-144*y1^4*y3-16*y2^4+84*y1*y2 ^2*y3+135*y1^2*y3^2+108*y1^3*y-36*y3^3+36*y2^2*y-54*y1*y3*y-81*y^2\}; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#,8*&\"#O\"\"\")%#y1G\"\"'F'F'*( \"$g\"F')F)\"\"$F')%#y2G\"\"#F'!\"\"*(\"$W\"F')F)\"\"%F'%#y3GF'F2*&\"# ;F')F0F6F'F2**\"#%)F'F)F'F/F'F7F'F'*(\"$N\"F')F)F1F')F7F1F'F'*(\"$3\"F 'F-F'%\"yGF'F'*&F&F')F7F.F'F2*(F&F'F/F'FCF'F'**\"#aF'F)F'F7F'FCF'F2*& \"#\")F')FCF1F'F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "# I o utput this as a set even though except for a pathological case, this s hould be a one element set" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 137 "schurf(composition) : R eturns schur function corresponding to the DELTA of composition with d istinct parts\nas if computed via slinky rule" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "schurf:=proc(p)\nlocal i,out,sg,siz,sp,d,la,n; \nn:=nops(p); \nsiz:=size(p);\nif siz " 0 "" {MPLTEXT 1 0 18 "schurf([3,8,5,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"sG6&\"\"&\"\"$\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 302 "findbas(GB,n,d) : Returns a monomial basis for the Quotient of Q[x1,...,x_n] / GB (Will omit basis elements of degree higher d so n eed to pick d high enough\n(Note that if Quotient finite dimensional t hen nops will stabilize as d grows, if Quotient not finite dim then no ps will grow bigger as d grows)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 430 "findbas:=proc(GB,n,d)\nlocal out, mons, mon, lmons, k , al,i, bool,te;\nal := [ seq(x||i, i = 1 .. n) ];\nte:= NULL;\nfor i from 1 t o nops(GB) do\nte := te , [leadmon(GB[i],al)][2];\nod;\nlmons:= [te]; \nout := NULL;\nfor k from 1 to d do\nmons:= getmons(n,k);\nfor mon in mons do\nbool:= true;\nfor i from 1 to nops(lmons) while bool do\nif \+ divide(mon,lmons[i]) then\nbool := false; fi;\nod;\nif bool then out \+ := out,mon;fi;\nod; od;\n[1, out];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "GTEST \+ := gbasis([evalsf(e1,x1+x2+x3)-y1,evalsf(e2,x1+x2+x3)-y2,evalsf(e3,x1+ x2+x3)-y3],[x1,x2,x3,y1,y2,y3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% >ESTG7%,*%#y3G!\"\"*&%#y2G\"\"\"%#x3GF+F+*$)F,\"\"$F+F+*&)F,\"\"#F+% #y1GF+F(,.*$)%#x2GF2F+F+*&F,F+F7F+F+*$F1F+F+*&F7F+F3F+F(*&F,F+F3F+F(F* F+,*%#x1GF+F7F+F,F+F3F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " FTEST := findbas(GTEST,3,10); nops(FTEST);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "# We do indeed get Artin Monomials." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&FTESTG7(\"\"\"%#x2G%#x3G*&F'F&F(F&*$)F(\"\"#F&*&F 'F&F+F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# See Part III R.1, part 8 of Qual for anothe r e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 52 "Procedures for Bases of Quotient Modulo " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 116 "geb(n) : Returns gbasis for [e1,...,en ] (ngeb(n) returns them for [en,...,e1])\ngbasis is more general mapl e command" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "geb:=proc(n)\n local al,out,ees,k,ee,sal,i;\nal:=[seq(x[n+1-i],i=1..n)];\nsal:=conver t(al,`+`);\nees:=NULL;\nfor k from 1 to n do\n ee:=evalsf(cat(e,k),sa l);\n ees:=ees,ee;\n od;\nout:=gbasis([ees],al,plex);\nend:\n \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "ngeb:=proc(n)\nlocal al, out,ees,k,ee,sal,i;\nal:=[seq(x[i],i=1..n)];\nsal:=convert(al,`+`);\ne es:=NULL;\nfor k from 1 to n do\n ee:=evalsf(cat(e,k),sal);\n ees:=e es,ee;\n od;\nout:=gbasis([ees],al);\nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "geb(2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,&&%\"xG6#\"\"#\"\"\"&F&6#F)F)*$ )F*F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "geb(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,(&%\"xG6#\"\"$\"\"\"&F&6#\"\"#F)&F&6#F)F) ,(*$)F*F,F)F)*&F*F)F-F)F)*$)F-F,F)F)*$)F-F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ngeb(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$) &%\"xG6#\"\"$F)\"\"\",(*$)&F'6#\"\"#F0F*F**&F.F*F&F*F**$)F&F0F*F*,(F&F *F.F*&F'6#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "map(print ,ngeb(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)&%\"xG6#\"\"$F(\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)&%\"xG6#\"\"#F)\"\"\"F**&F&F* &F'6#\"\"$F*F**$)F,F)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(&%\"xG6 #\"\"$\"\"\"&F%6#\"\"#F(&F%6#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "# gbasis is a more ge neral maple command, for help see ?gbasis" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 157 "norma(P, n) : Returns P modulo ideal of gbasis [e1,...,en] using plex order (nn orma(P,n) returns them modulo [en,...e1])\nnormalf is more general map le command" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "norma:=proc(P ,n)\nlocal al,i,out:\nal:=[seq(x[n+1-i],i=1..n)]; \nout:=normalf(P,g eb(n),al,plex);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "nn orma:=proc(P,n)\nlocal al,i,out;\nal:=[seq(x[n+1-i],i=1..n)]; \nout: =normalf(P,ngeb(n),al);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "norma(x[1]*x[2]^2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&)&% \"xG6#\"\"\"\"\"#F)&F'6#F*F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "al := [ seq(x[3+1-m],m=1..3)]; \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#alG7%&%\"xG6#\"\"$&F'6#\"\"#&F'6#\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "geb(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,(&%\"xG6#\"\"$\"\"\"&F&6#\"\"#F)&F&6#F)F),(*$)F*F,F) F)*&F*F)F-F)F)*$)F-F,F)F)*$)F-F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "x[1]*(x[2]^2 + x[2]*x[1])+x[1]^3; # == 0 since in th e ideal, thus x[1]*x[2]^2 == - x[2]*x[1]^2 + x[1]^3 == - x[2]*x[1]^2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%\"xG6#\"\"\"F(,&*$)&F&6#\"\"#F .F(F(*&F,F(F%F(F(F(F(*$)F%\"\"$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "expand( x[1]*(x[2]^2 + x[2]*x[1])+x[1]^3) - x[1]^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%\"xG6#\"\"\"F()&F&6#\"\"#F,F(F(*&)F%F,F(F*F(F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "nnorma(x[1]*x[2]^2,3); # T his monomial already fully reduced using other gbasis" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"xG6#\"\"\"F')&F%6#\"\"#F+F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ngeb(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$)&%\"x G6#\"\"$F)\"\"\",(*$)&F'6#\"\"#F0F*F**&F.F*F&F*F**$)F&F0F*F*,(F&F*F.F* &F'6#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "# \"normalf\" \+ is built-in maple command, see ?normalf for help" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "fix pol(P,n) : Returns P modulo " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 220 "fixpol:=proc(P,n)\nlocal AL,SAL,GB,BAS,out,i;\nopt ion remember:\nAL:=[seq(x[i],i=1..n)];\nSAL:=size(AL);\nBAS:=map(expan d,(map(evalsf,[seq(cat(e,i),i=1..n)],SAL)));print(BAS);\nGB:=gbasis(BA S,AL); \nout:=normalf(P,GB,AL);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fixpol( x[2]^2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#&%\"xG6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$)&%\"xG6#\"\"#F(\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fixpol(x[1]*x[2],3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%,(&%\"xG6#\"\"$\"\"\"&F&6#\"\"#F)&F&6#F)F),(*& F*F)F%F)F)*&F-F)F%F)F)*&F*F)F-F)F)*(F%F)F*F)F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)&%\"xG6#\"\"$\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 84 "mkgram(B, n) : Returns the Gramm Matrix for a given basis B in n vars (WHAT IS T HIS?)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "mkgram:=proc(B,n) \nlocal i,mat,j,N,rw;\nmat:=NULL;\nN:=nops(B);\nfor i from 1 to N do\n rw:=NULL;\n for j from 1 to N do\n rw:=rw,scalPQ(B[i],B[j],n); \n od;\n mat:=mat,[rw];\n od;\nmatrix([mat] );\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Goo:=mkgram(art (3),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GooGK%'matrixG6#7(7(\"\" !F*F*F*F*\"\"\"7(F*F*F*!\"\"F+F*7(F*F*F*F+F*&%\"sG6#F+7(F*F-F+F*F/&F06 $F+F+7(F*F+F*F/F*&F06#\"\"#7(F+F*F/F3F6&F06$F8F+Q*pprint1996\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "IG:=inverse(Goo);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IGG K%'matrixG6#7(7(,*&%\"sG6$\"\"#\"\"\"!\"\"*(F.F/&F,6#F/F/&F,6#F.F/F/*( F.F/F2F/&F,6$F/F/F/F/*&F.F/)F2\"\"$F/F0,&F4F0*$)F2F.F/F/,(F4F0*&F.F/F> F/F/F7F0,$F2F0FAF/7(F<\"\"!FAFCF/FC7(F?FA,$*&F.F/F2F/F0F/F/FC7(FAFCF/F CFCFC7(FAF/F/FCFCFC7(F/FCFCFCFCFCQ*pprint2006\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "BB3:=multiply([art(3)],IG);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$BB3GK%'matrixG6#7#7(,4&%\"sG6$\"\"#\"\"\"!\"\"*(F. F/&F,6#F/F/&F,6#F.F/F/*(F.F/F2F/&F,6$F/F/F/F/*&F.F/)F2\"\"$F/F0*&&%\"x GF5F/,&F4F0*$)F2F.F/F/F/F/*&&F>6#F;F/,(F4F0*&F.F/FAF/F/F7F0F/F/*(F=F/F CF/F2F/F0*&)FCF.F/F2F/F0*&F=F/FIF/F/,*F4F0F@F/*&FCF/F2F/F0*$FIF/F/,0F4 F0*&F.F/FAF/F/F7F0*&F=F/F2F/F0*(F.F/FCF/F2F/F0*&F=F/FCF/F/FMF/,&F2F0FC F/,(F2F0F=F/FCF/F/Q*pprint2016\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "MandM := map(expand,map(evalsf,BB3,x[1]+x[2]+x[3])); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&MandMGK%'matrixG6#7#7(,$*&&%\"x G6#\"\"#\"\"\")&F-6#F0F/F0!\"\"*&F,F0F2F0,&F5F0*$F1F0F0,&F,F4F2F4,$F2F 4F0Q*pprint2026\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 83 "mkdual(alpha) : Returns the dual \+ to the artin basis (UNCLEAR HOW DEPENDS ON ALPHA)" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 361 "mkdual:=proc(alpha)\nlocal i,beta,out,n,te, pal,j,sg;\nn:=nops(alpha);\nbeta:=[seq(0,i=1..n)];\nfor i from 1 to n \+ do\n beta[i]:=i-1-alpha[i];\n od;\nsg:=(-1)^convert(beta,`+`);\no ut:=1;\nfor i from 2 to n do\n pal:=convert([seq(x[j],j=1..i-1)],`+` );#print(i,pal);\n\n te:=evalsf(cat(e,beta[i]),pal);#print(te);\n\n \+ out:=out*te;\n od;\nsubs(e0=1,sg*out);\nend:\n \n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "DU := mkdual([0,1,1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DUG*&,&&%\"xG6#\"\"#\"\"\"&F(6#F+F+F+,(&F (6#\"\"$F+F'F+F,F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "exp and(DU);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"xG6#\"\"#\"\"\"&F& 6#\"\"$F)F)*$)F%F(F)F)*(F(F)F%F)&F&6#F)F)F)*&F0F)F*F)F)*$)F0F(F)F)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 97 "expandit(P,n) : Returns P expanded in Artin/Symm basis \+ assuming n vars (Shows intermediate steps)" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 249 "expandit:=proc(P,n)\nlocal expos,out,q,te,bb,arts; \narts:=art(n);print(arts);print(arts[3]);\nexpos:=map(getexpo,art(n), n);\nout:=0;\nfor q in expos do\nbb:=mkdual(q);print(bb);\n te:=expand (scalPQ(bb,P,n));print(te);\n out:=out+te*mkmon(q);;\n od;\nout;\nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "art(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"&%\"xG6#\"\"#&F&6#\"\"$*&F%F$F)F$*$)F)F(F$ *&F%F$F.F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "art(3)[2];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"xG6#\"\"#" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "expandit(x[1]^5,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"&%\"xG6#\"\"#&F&6#\"\"$*&F%F$F)F$*$)F)F(F$*&F%F $F.F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"xG6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&&%\"xG6#\"\"\"F(,(*&#F(\"\"#F(*$)&F&6#F,F,F(F( !\"\"*&#F(F,F(*$)F%F,F(F(F1*&#F(F,F(*$),&F/F(F%F(F,F(F(F(F(F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"sG6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"#F&*$)&%\"xG6#F'F'F&F&!\"\"*&#F&F'F&*$)& F+6#F&F'F&F&F-*&#F&F'F&*$),&F*F&F2F&F'F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$&%\"sG6#\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&&%\"xG6#\"\"\"F',&&F%6#\"\"#F'F$F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$&%\"sG6#\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"xG 6#\"\"#!\"\"&F%6#\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"sG6# \"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$&%\"xG6#\"\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,*&%\"sG6#\"\"&\"\"\"*&&F%6#\"\"%F(&%\"xG6#\"\"#F(!\" \"*&F*F(&F.6#\"\"$F(F1*(&F%F4F(F-F(F3F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 73 "ToFact := evalsf(s[5]-s[4]*x[2]-s[4]*x[3]+s[3]*x[3] *x[2],x[1]+x[2]+x[3]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'ToFactG,D*&#\"\"\"\"$?\"F(*$),(&%\"xG6#\" \"$F(&F.6#\"\"#F(&F.6#F(F(\"\"&F(F(F(*&,&*&#F(\"#CF(F1F(!\"\"*&#F(F;F( F-F(F " 0 "" {MPLTEXT 1 0 15 "factor(ToFact);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*$)&%\"xG6#\"\"\"\"\"&F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "expandit(3,3); # Thinking of 3 as a monomial" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"&%\"xG6#\"\"#&F&6#\"\"$*&F%F$F )F$*$)F)F(F$*&F%F$F.F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"xG6#\"\" $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&&%\"xG6#\"\"\"F(,(*&#F(\"\"#F (*$)&F&6#F,F,F(F(!\"\"*&#F(F,F(*$)F%F,F(F(F1*&#F(F,F(*$),&F/F(F%F(F,F( F(F(F(F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"#F&*$)&%\"xG6#F'F'F&F&!\"\"*&#F&F'F&*$)& F+6#F&F'F&F&F-*&#F&F'F&*$),&F*F&F2F&F'F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"xG6#\"\" \"F',&&F%6#\"\"#F'F$F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"xG6#\"\"#!\"\"&F%6#\"\"\"F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$&%\"xG6#\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Old Qual Questions" }} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "R2-2002" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "D6:=mkdihedr(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#D6G7.K%'matrixG6#7$7$\"\"\"\"\"!7$F,F+Q*pprint2036\"KF'6#7$7$#F+\" \"#,$*&F5!\"\"\"\"$F4F87$,$*&F5F8F9F4F+F4Q*pprint204F/KF'6#7$7$#F8F5F6 7$F;FBQ*pprint205F/KF'6#7$7$F8F,7$F,F8Q*pprint206F/KF'6#7$7$FBF;7$F6FB Q*pprint207F/KF'6#7$7$F4F;7$F6F4Q*pprint208F/KF'6#7$F*FIQ*pprint209F/K F'6#7$F3FOQ*pprint210F/KF'6#7$FAFUQ*pprint211F/KF'6#7$FHF-Q*pprint212F /KF'6#7$FNF:Q*pprint213F/KF'6#7$FTFCQ*pprint214F/" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 83 "# D6 is not abelian so a faithful representa tion of dimension 2 will be irreducible" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "A := D6 [2]; B := D6[12]; multiply(A,B)- multiply(B,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGK%'matrixG6#7$7$#\"\"\"\"\"#,$*&F,!\"\"\"\"$F*F/7 $,$*&F,F/F0F*F+F*Q*pprint2156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"BGK%'matrixG6#7$7$#\"\"\"\"\"#,$*&F,!\"\"\"\"$F*F+7$F-#F/F,Q*pprint2 166\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&K%'matrixG6#7$7$#!\"\"\"\"# ,$*&F+F*\"\"$#\"\"\"F+F07$F,F/Q*pprint2176\"F0KF%6#7$7$F0\"\"!7$F8F*Q* pprint218F3F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "for j fro m 1 to 12 do; C[j] := NULL; for i from 1 to 12 do; C[j] := C[j] union \+ \{multiply((D6[i],D6[j] ),inverse(D6[i]) )\}; od; print(C[j]); od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"\"6\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<.K%'matrixG6#7$7$\"\"\"\"\"!7$F*F)Q*pprint2246\"KF%F &Q*pprint225F-KF%F&Q*pprint226F-KF%F&Q*pprint227F-KF%F&Q*pprint230F-KF %F&Q*pprint222F-KF%F&Q*pprint223F-KF%F&Q*pprint219F-KF%F&Q*pprint220F- KF%F&Q*pprint221F-KF%F&Q*pprint228F-KF%F&Q*pprint229F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"#6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.K%'matrixG6#7$7$#\"\"\"\"\"#,$*&F+!\"\"\"\"$F)F*7$,$*&F+F.F/F)F.F )Q*pprint2316\"KF%F&Q*pprint232F4KF%6#7$7$F)F17$F,F)Q*pprint233F4KF%F& Q*pprint234F4KF%F8Q*pprint242F4KF%F8Q*pprint235F4KF%F8Q*pprint236F4KF% F&Q*pprint237F4KF%F&Q*pprint238F4KF%F&Q*pprint239F4KF%F8Q*pprint240F4K F%F8Q*pprint241F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"$6\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.K%'matrixG6#7$7$#!\"\"\"\"#,$*&F +F*\"\"$#\"\"\"F+F07$,$*&F+F*F.F/F*F)Q*pprint2476\"KF%6#7$7$F)F27$F,F) Q*pprint248F5KF%F&Q*pprint249F5KF%F&Q*pprint250F5KF%F7Q*pprint251F5KF% F7Q*pprint252F5KF%F&Q*pprint253F5KF%F7Q*pprint254F5KF%F&Q*pprint243F5K F%F7Q*pprint244F5KF%F&Q*pprint245F5KF%F7Q*pprint246F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"%6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.K%'matrixG6#7$7$!\"\"\"\"!7$F*F)Q*pprint2556\"KF%F&Q*pprint256F-K F%F&Q*pprint257F-KF%F&Q*pprint258F-KF%F&Q*pprint264F-KF%F&Q*pprint259F -KF%F&Q*pprint260F-KF%F&Q*pprint261F-KF%F&Q*pprint262F-KF%F&Q*pprint26 3F-KF%F&Q*pprint265F-KF%F&Q*pprint266F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"&6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.K%'matrix G6#7$7$#!\"\"\"\"#,$*&F+F*\"\"$#\"\"\"F+F07$,$*&F+F*F.F/F*F)Q*pprint27 36\"KF%F&Q*pprint274F5KF%6#7$7$F)F27$F,F)Q*pprint275F5KF%F&Q*pprint276 F5KF%F9Q*pprint277F5KF%F&Q*pprint268F5KF%F9Q*pprint269F5KF%F9Q*pprint2 70F5KF%F9Q*pprint271F5KF%F9Q*pprint272F5KF%F&Q*pprint267F5KF%F&Q*pprin t278F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"'6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.K%'matrixG6#7$7$#\"\"\"\"\"#,$*&F+!\"\"\" \"$F)F.7$,$*&F+F.F/F)F*F)Q*pprint2836\"KF%F&Q*pprint284F4KF%6#7$7$F)F1 7$F,F)Q*pprint288F4KF%F8Q*pprint289F4KF%F8Q*pprint290F4KF%F8Q*pprint28 5F4KF%F&Q*pprint282F4KF%F&Q*pprint286F4KF%F&Q*pprint279F4KF%F8Q*pprint 280F4KF%F8Q*pprint287F4KF%F&Q*pprint281F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<. K%'matrixG6#7$7$#!\"\"\"\"#,$*&F+F*\"\"$#\"\"\"F+F*7$F,F/Q*pprint3016 \"KF%6#7$7$F0\"\"!7$F8F*Q*pprint297F3KF%F5Q*pprint302F3KF%F5Q*pprint29 2F3KF%6#7$7$F),$*&F+F*F.F/F07$FCF/Q*pprint298F3KF%F@Q*pprint291F3KF%F@ Q*pprint299F3KF%F5Q*pprint293F3KF%F@Q*pprint294F3KF%F&Q*pprint295F3KF% F&Q*pprint300F3KF%F&Q*pprint296F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%\"CG6#\"\")6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.K%'matrixG6#7$7$ !\"\"\"\"!7$F*\"\"\"Q*pprint3126\"KF%6#7$7$#F,\"\"#,$*&F4F)\"\"$F3F,7$ F5#F)F4Q*pprint313F.KF%6#7$7$F3,$*&F4F)F7F3F)7$F?F9Q*pprint304F.KF%F&%\"CG6#\"\"*6\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.K%'matrixG6#7$7$#!\"\"\"\"#,$*&F +F*\"\"$#\"\"\"F+F07$F,F/Q*pprint3266\"KF%6#7$7$F0\"\"!7$F8F*Q*pprint3 22F3KF%6#7$7$F),$*&F+F*F.F/F*7$F?F/Q*pprint320F3KF%F5Q*pprint324F3KF%F &%\"CG6#\"#56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.K%'matrixG6#7$7$!\"\"\"\"!7$F*\"\"\"Q*pprint3296\" KF%6#7$7$#F,\"\"#,$*&F4F)\"\"$F3F,7$F5#F)F4Q*pprint334F.KF%F&Q*pprint3 31F.KF%F&Q*pprint335F.KF%F0Q*pprint332F.KF%6#7$7$F3,$*&F4F)F7F3F)7$FEF 9Q*pprint337F.KF%FBQ*pprint330F.KF%F&Q*pprint336F.KF%F0Q*pprint333F.KF %FBQ*pprint327F.KF%FBQ*pprint338F.KF%F0Q*pprint328F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"#66\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# <.K%'matrixG6#7$7$#!\"\"\"\"#,$*&F+F*\"\"$#\"\"\"F+F07$F,F/Q*pprint342 6\"KF%F&Q*pprint343F3KF%6#7$7$F),$*&F+F*F.F/F*7$F:F/Q*pprint344F3KF%F& Q*pprint345F3KF%F7Q*pprint339F3KF%6#7$7$F0\"\"!7$FFF*Q*pprint346F3KF%F &Q*pprint347F3KF%FCQ*pprint340F3KF%F7Q*pprint341F3KF%FCQ*pprint348F3KF %FCQ*pprint349F3KF%F7Q*pprint350F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"CG6#\"#76\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.K%'matrixG6#7$7$ !\"\"\"\"!7$F*\"\"\"Q*pprint3596\"KF%6#7$7$#F,\"\"#,$*&F4F)\"\"$F3F,7$ F5#F)F4Q*pprint360F.KF%F&Q*pprint357F.KF%F0Q*pprint358F.KF%6#7$7$F3,$* &F4F)F7F3F)7$FCF9Q*pprint353F.KF%F@Q*pprint354F.KF%F&Q*pprint362F.KF%F &Q*pprint351F.KF%F@Q*pprint352F.KF%F@Q*pprint355F.KF%F0Q*pprint356F.KF %F0Q*pprint361F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "# Conj ugacy Classes with repeats = \{D6[1]\}, \{D6[2],D6[6]\}, \{D6[3],D6[5] \}, \{D6[4]\}, \{D6[7],D6[9],D6[11]\}, D6[8]\} , \{D6[8], D6[10],D6[12 ]\} = \{C1, C2, C3, C4, C7, C8\} " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "COM:=\{\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "for i to 12 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " for j to 12 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 " COM:=COM union \{multiply(D6[i],D6[j],inverse(D6[i]),inverse(D6 [j]))\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 3 "od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "COM; # Commutator Subgroup" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#<\\tK%'matrixG6#7$7$#!\"\"\"\"#,$*&F+F*\"\"$#\"\"\"F+ F07$,$*&F+F*F.F/F*F)Q*pprint4106\"KF%6#7$7$F)F27$F,F)Q*pprint411F5KF%6 #7$7$F0\"\"!7$F@F0Q*pprint412F5KF%F&Q*pprint413F5KF%F7Q*pprint414F5KF% F&Q*pprint405F5KF%F7Q*pprint406F5KF%F7Q*pprint407F5KF%F7Q*pprint408F5K F%F=Q*pprint409F5KF%F=Q*pprint456F5KF%F=Q*pprint457F5KF%F=Q*pprint458F 5KF%F&Q*pprint459F5KF%F7Q*pprint363F5KF%F7Q*pprint364F5KF%F&Q*pprint36 7F5KF%F&Q*pprint368F5KF%F7Q*pprint369F5KF%F&Q*pprint370F5KF%F7Q*pprint 371F5KF%F&Q*pprint385F5KF%F&Q*pprint386F5KF%F=Q*pprint387F5KF%F=Q*ppri nt388F5KF%F7Q*pprint389F5KF%F=Q*pprint379F5KF%F7Q*pprint380F5KF%F=Q*pp rint381F5KF%F=Q*pprint382F5KF%F7Q*pprint383F5KF%F=Q*pprint469F5KF%F=Q* pprint470F5KF%F&Q*pprint471F5KF%F=Q*pprint472F5KF%F=Q*pprint473F5KF%F7 Q*pprint474F5KF%F=Q*pprint460F5KF%F&Q*pprint461F5KF%F7Q*pprint467F5KF% F&Q*pprint468F5KF%F=Q*pprint395F5KF%F7Q*pprint396F5KF%F=Q*pprint397F5K F%F=Q*pprint398F5KF%F=Q*pprint399F5KF%F=Q*pprint390F5KF%F7Q*pprint391F 5KF%F7Q*pprint392F5KF%F=Q*pprint393F5KF%F&Q*pprint394F5KF%F=Q*pprint37 8F5KF%F=Q*pprint384F5KF%F&Q*pprint502F5KF%F=Q*pprint503F5KF%F7Q*pprint 504F5KF%F=Q*pprint505F5KF%F&Q*pprint506F5KF%F7Q*pprint478F5KF%F&Q*ppri nt479F5KF%F=Q*pprint480F5KF%F=Q*pprint481F5KF%F7Q*pprint482F5KF%F=Q*pp rint487F5KF%F&Q*pprint488F5KF%F=Q*pprint489F5KF%F=Q*pprint490F5KF%F=Q* pprint491F5KF%F7Q*pprint377F5KF%F=Q*pprint365F5KF%F&Q*pprint366F5KF%F= Q*pprint372F5KF%F=Q*pprint373F5KF%F=Q*pprint374F5KF%F&Q*pprint375F5KF% F=Q*pprint376F5KF%F&Q*pprint400F5KF%F7Q*pprint401F5KF%F=Q*pprint402F5K F%F=Q*pprint403F5KF%F&Q*pprint404F5KF%F=Q*pprint431F5KF%F7Q*pprint432F 5KF%F&Q*pprint433F5KF%F=Q*pprint434F5KF%F7Q*pprint435F5KF%F7Q*pprint44 6F5KF%F7Q*pprint447F5KF%F=Q*pprint448F5KF%F&Q*pprint449F5KF%F7Q*pprint 450F5KF%F=Q*pprint441F5KF%F&Q*pprint442F5KF%F&Q*pprint443F5KF%F=Q*ppri nt444F5KF%F=Q*pprint445F5KF%F7Q*pprint426F5KF%F&Q*pprint427F5KF%F=Q*pp rint428F5KF%F=Q*pprint429F5KF%F&Q*pprint430F5KF%F=Q*pprint420F5KF%F7Q* pprint421F5KF%F=Q*pprint422F5KF%F&Q*pprint423F5KF%F=Q*pprint424F5KF%F7 Q*pprint425F5KF%F=Q*pprint415F5KF%F=Q*pprint416F5KF%F7Q*pprint417F5KF% F=Q*pprint418F5KF%F7Q*pprint419F5KF%F=Q*pprint436F5KF%F=Q*pprint437F5K F%F&Q*pprint438F5KF%F&Q*pprint439F5KF%F&Q*pprint440F5KF%F&Q*pprint483F 5KF%F=Q*pprint484F5KF%F=Q*pprint485F5KF%F&Q*pprint486F5KF%F7Q*pprint45 1F5KF%F&Q*pprint452F5KF%F7Q*pprint453F5KF%F&Q*pprint454F5KF%F=Q*pprint 455F5KF%F=Q*pprint492F5KF%F=Q*pprint493F5KF%F=Q*pprint494F5KF%F&Q*ppri nt495F5KF%F=Q*pprint496F5KF%F=Q*pprint475F5KF%F=Q*pprint476F5KF%F7Q*pp rint477F5KF%F=Q*pprint497F5KF%F7Q*pprint498F5KF%F=Q*pprint499F5KF%F=Q* pprint500F5KF%F=Q*pprint501F5KF%F=Q*pprint462F5KF%F=Q*pprint463F5KF%F= Q*pprint464F5KF%F=Q*pprint465F5KF%F=Q*pprint466F5" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "# Hence | G / COM | = 4 which implies 4 one-dimsional representat ions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "# Further, 12 = 1^2 + 1^2 + 1^2 + 1^2 + + 2^2 + X implies X = 2^2 and thus there is another two-dimensional \+ representation # besides given." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "R4-2002" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 137 "# Notice there are two methods for creating a gro up algebra element of C[S_n]. In mkgel, the numbers in front are the \+ # coefficients. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "F:=mkgel([[1,[2,3,4,5,1]],[1 ,[1,3,4,5,2]],[1,[1,2,4,5,3]],[1,[1,2,3,5,4]],[1,[1,2,3,4,5]]]);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "F2 := xsig([2,3,4,5,1])+xsig([1,3,4 ,5,2])+xsig([1,2,4,5,3])+xsig([1,2,3,5,4])+xsig([1,2,3,4,5]);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,,&%\"xG6'\"\"#\"\"$\"\"%\"\"&\" \"\"F-&F'6'F-F*F+F,F)F-&F'6'F-F)F+F,F*F-&F'6'F-F)F*F,F+F-&F'6'F-F)F*F+ F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#F2G,,&%\"xG6'\"\"#\"\"$\"\" %\"\"&\"\"\"F-&F'6'F-F*F+F,F)F-&F'6'F-F)F+F,F*F-&F'6'F-F)F*F,F+F-&F'6' F-F)F*F+F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "FT:=fourier (F,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FTG7)7$7#\"\"&K%'matrixG6 #7#F'Q*pprint5076\"7$7$\"\"%\"\"\"KF*6#7&7&\"\"$\"\"!F8F87&\"\"#F:F8F8 7&F8F7F2F87&F8F8F1F8Q*pprint508F.7$7$F7F:KF*6#7'7'F2F8F8F8F87'F:F8F8F8 F87'F8!\"#F2F2F87'F:FFF2F2F87'F8!\"\"F7FIF2Q*pprint509F.7$7%F7F2F2KF*6 #7(7(F:F8F2F8F8F87(F2F2F2F8F8F87(F8F:F2F8F8F87(F:F8FIF2F2F87(F8F:FIF2F 2F87(F8F8F8F7FIF2Q*pprint510F.7$7%F:F:F2KF*6#7'7'F8F8F2F8F87'FIF2F2F8F 8Ffn7'FFF8FIF2F27'F8FFF2F2F2Q*pprint511F.7$7&F:F2F2F2KF*6#7&7&F2F8F8F8 7&F2F2FIF8F`o7&F:FIF2F8Q*pprint512F.7$7'F2F2F2F2F2KF*6#7#7#F2Q*pprint5 13F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "DD:=diag(FT[1,2],FT [2,2],FT[3,2],FT[4,2],FT[5,2],FT[6,2],FT[7,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DDGK%'matrixG6#7<7<\"\"&\"\"!F+F+F+F+F+F+F+F+F+F+F+F +F+F+F+F+F+F+F+F+F+F+F+F+7 " 0 "" {MPLTEXT 1 0 14 "eigenvals(DD);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6<\"\"&\"\"$F$\"\"#F%F%F%\"\"!F&F&F&F&F&F&F&F&\"\"\"F'F'F'F'F'F'F'F'F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factor(minpoly(DD,t)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,%\"tG\"\"\",&F$F%\"\"&!\"\"F%,&F $F%F%F(F%,&F$F%\"\"#F(F%,&F$F%\"\"$F(F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "S1-2002" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "unassign('y1', 'y2', 'y3'); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "elem5 := toe(p5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&elem5G,0*&\"\"&\"\"\"%#e5GF(F(*(F'F (%#e1GF(%#e4GF(!\"\"*(F'F(%#e3GF(%#e2GF(F-*(F'F(F/F()F+\"\"#F(F(*(F'F( )F0F3F(F+F(F(*(F'F(F0F()F+\"\"$F(F-*$)F+F'F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 54 "In3 := subs(e4=0,e5=0,elem5);\neval(elem5,[e4= 0,e5=0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$In3G,,*(\"\"&\"\"\"%#e 3GF(%#e2GF(!\"\"*(F'F(F)F()%#e1G\"\"#F(F(*(F'F()F*F/F(F.F(F(*(F'F(F*F( )F.\"\"$F(F+*$)F.F'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(\"\"&\" \"\"%#e3GF&%#e2GF&!\"\"*(F%F&F'F&)%#e1G\"\"#F&F&*(F%F&)F(F-F&F,F&F&*(F %F&F(F&)F,\"\"$F&F)*$)F,F%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "Po5 := top(In3);\n# The power symetric funtion P5 written in terms of P1, P2, P3 assuming three variables" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Po5G,**&#\"\"\"\"\"'F(*$)%#p1G\"\"&F(F(F(*&#F-F)F(*& %#p2GF()F,\"\"$F(F(!\"\"*&#F-F)F(*&%#p3GF()F,\"\"#F(F(F(*&F6F(*&F8F(F1 F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "P5 := simplify(evalsf(Po5,x1+x2+x3));\n# \+ Check that this expression is indeed P5 in three variables" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P5G,(*$)%#x2G\"\"&\"\"\"F**$)%#x3GF)F*F** $)%#x1GF)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "P1:=evalsf (p1,x1+x2+x3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G,(%#x1G\"\" \"%#x2GF'%#x3GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P2:=eva lsf(p2,x1+x2+x3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P2G,(*$)%#x1G \"\"#\"\"\"F**$)%#x2GF)F*F**$)%#x3GF)F*F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "P3:=evalsf(p3,x1+x2+x3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P3G,(*$)%#x1G\"\"$\"\"\"F**$)%#x2GF)F*F**$)%#x3GF)F* F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "GB := gbasis([P1-y1,P2-y2,P3-y3],[x1,x2,x3,y 1,y2,y3],plex); # plex species the ordering for the monomials" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GBG7%,*%#x1G\"\"\"%#x2GF(%#x3GF(%#y 1G!\"\",0*&\"\"#F()F)F/F(F(*&F/F()F*F/F(F(%#y2GF,*(F/F(F)F(F*F(F(*(F/F (F)F(F+F(F,*(F/F(F*F(F+F(F,*$)F+F/F(F(,0*&F/F(%#y3GF(F,*(\"\"$F(F*F(F8 F(F(*$)F+F=F(F,*&\"\"'F()F*F=F(F(*(F=F(F3F(F*F(F,*(FAF(F2F(F+F(F,*(F=F (F3F(F+F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "normalf(P5,GB,[x1,x2,x3,y1,y2,y3],p lex);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"'F&*$)%#y1G\" \"&F&F&F&*&#F+F'F&*&)F*\"\"#F&%#y3GF&F&F&*&#F+F'F&*&)F*\"\"$F&%#y2GF&F &!\"\"*&F-F&*&F7F&F1F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "# Since this expression consists of only y's (no x's) that means Po5 is in the ideal using that alphabet. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "# testAI will also do the trick" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "testAI(P5, [P1,P2,P3],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"'F&*$)%#y1G\"\"&F&F&F&*&#F+F'F&*&)F*\"\" #F&%#y3GF&F&F&*&#F+F'F&*&)F*\"\"$F&%#y2GF&F&!\"\"*&F-F&*&F7F&F1F&F&F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 7 "S2-2002" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sk:=skew(s[3,2],s[4,4,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s kG,6*(\"\")!\"\"%#p4G\"\"\"%#p1G\"\"#F(*(\"#;F(%#p2GF,F+F,F(*(F'F(F)F* F/F*F**&\"#=F(%#p3GF,F(**\"\"'F(F3F*F+F*F/F*F(*(F2F(F3F*F+\"\"$F(*&F5F (%#p6GF*F**(\"\"(F*\"$W\"F(F+F5F***\"\"&F*\"#[F(F/F*F+\"\"%F**&F?F(F/F 7F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "tos(sk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(&%\"sG6$\"\"%\"\"#\"\"\"&F%6%F'F)F)F)&F%6%\" \"$F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "toe(sk);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&%#e4G\"\"\")%#e1G\"\"#F&!\"\"*&)%# e2GF)F&F'F&F&*(%#e3GF&F(F&F-F&F**&F(F&%#e5GF&F&*$)F-\"\"$F&F**(F)F&F%F &F-F&F&%#e6GF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "tom(sk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4&%\"mG6$\"\"%\"\"#\"\"\"&F%6$\"\"$ F,F)*&\"\")F)&F%6&F,F)F)F)F)F)*&F'F)&F%6%F,F(F)F)F)*&F(F)&F%6%F'F)F)F) F)*&\"\"'F)&F%6%F(F(F(F)F)*&\"#6F)&F%6&F(F(F)F)F)F)*&\"#?F)&F%6'F(F)F) F)F)F)F)*&\"#NF)&F%6(F)F)F)F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 346 "LittleRichardson := matrix([[h3,h5,h6],[1,h2,h3],[0, 1,h1]]);\n\n# We can also compute the symmetric funtion corresponding to a skew tableau by taking the determinant of a matrix of # homoge neous functions\n# Here the indicies of the top row are [3,4,4]+[0,1, 2] and indicies of the first column are [3,2,0] - [0,1,2] where h1 = 1 # and h_-n = 0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1LittleRichards onGK%'matrixG6#7%7%%#h3G%#h5G%#h6G7%\"\"\"%#h2GF*7%\"\"!F.%#h1GQ*pprin t5156\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "det(LittleRichardson);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%#h1G\"\"\"%#h5GF&!\"\"*(%#h3GF&F%F&%#h2GF& F&*$)F*\"\"#F&F(%#h6GF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "t oh(sk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%#h1G\"\"\"%#h5GF&!\"\" *(%#h3GF&F%F&%#h2GF&F&*$)F*\"\"#F&F(%#h6GF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "tos(sk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(&%\"s G6$\"\"%\"\"#\"\"\"&F%6%F'F)F)F)&F%6%\"\"$F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "# There is also a way to find this schur fu nction by hand by finding what kinds of tableau can be created by thes e skew # hooks." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "S3-2002" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "S6:=permute(6): # Create the symmetric group S_6. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 89 "mkpowsym([1,1,3]); # Recall how mkpowsym creat es power symmetric function from a sequence" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#p3G\"\"\")%#p1G\"\"#F%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "prep ([2,2,2]): trace(NAT(S6[56]));\nchilasig([2,2,2],S6[56]); # Two ways o f computing the irreducible character corresponding to [2,2,2] on 56th permutation of S_6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "ans:=0 : for i to 720 do ans:=ans + chilasig([2,2,2],S6[i])*chilasig([2,2,2], S6[i])*mkpowsym(cycstr(S6[i])); od: ans:=ans/720;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ansG,4*(\"\"&\"\"\"\"$W\"!\"\"%#p1G\"\"'F(*(\"#[F*%# p2GF(F+\"\"%F(*(\"#=F*%#p3GF(F+\"\"$F(*(\"#;F*F/\"\"#F+F7F(*(\"\")F*%# p4GF(F+F7F(**F,F*F3F(F+F(F/F(F(*(F4F(F6F*F/F4F(*(F9F*F:F(F/F(F(*(F7F( \"\"*F*F3F7F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tos(ans);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*&%\"sG6#\"\"'\"\"\"&F%6$\"\"%\"\"# F(&F%6&\"\"$F(F(F(F(&F%6%F,F,F,F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# So repn of S_6 corresponding to A[2,2,2] tensor A[2 ,2,2] decomposes into" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "# \+ A[6] + A[4,2] + A[3,1,1,1] + A[2,2,2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "# Usin g the SF package's command (itensor) that computes the decomposition o f (inner) tensors, we could have decomposed this immediately." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 60 "tos(itensor(s[2,2,2],s[2,2,2])); itensor(s[2,2,2],s [2,2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*&%\"sG6#\"\"'\"\"\"&F%6 $\"\"%\"\"#F(&F%6&\"\"$F(F(F(F(&F%6%F,F,F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*(\"\"&\"\"\"\"$W\"!\"\"%#p1G\"\"'F&*(\"#[F(%#p2GF&F) \"\"%F&*(\"#=F(%#p3GF&F)\"\"$F&*(\"#;F(F-\"\"#F)F5F&*(\"\")F(%#p4GF&F) F5F&**F*F(F1F&F)F&F-F&F&*(F2F&F4F(F-F2F&*(F7F(F8F&F-F&F&*(F5F&\"\"*F(F 1F5F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "# So answer is th e power symmetric function which is the Frobenius image of the rep'n A [6] + A[4,2] + A[3,1,1,1] + A[2,2,2] " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "S3 2000" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Tensored := itensor(itensor( s[3,2,1],s[3,2,1]),s[4,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Tens oredG,&*&#\"\"\"\"\"&F(*&%#p5GF(%#p1GF(F(!\"\"*&#\"#;F)F(*$)F,\"\"'F(F (F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tos(Tensored);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,8*&\"\"$\"\"\"&%\"sG6#\"\"'F&F&*&\"#; F&&F(6$\"\"&F&F&F&*&\"#HF&&F(6$\"\"%\"\"#F&F&*&\"#KF&&F(6%F4F&F&F&F&*& F,F&&F(6$F%F%F&F&*&\"#^F&&F(6%F%F5F&F&F&*&F7F&&F(6&F%F&F&F&F&F&*&F,F&& F(6%F5F5F5F&F&*&F1F&&F(6&F5F5F&F&F&F&*&F,F&&F(6'F5F&F&F&F&F&F&*&F%F&&F (6(F&F&F&F&F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "# \+ This is the expansion of A[3,2,1] tensor A[3,2,1] tensor A[4,2] into i rreducible representations of S6 (written out as a symmetric function) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "# Since in this sum w e have one instead of a p[lambda] term, we set p1,p5,p6 to be one to f ind the corresponding integer." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eval(Tensored,[p 1=1,p5=1,p6=1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "# This integer is (1/n!) SUM_(perm s of S_n) Chi^B(sig) where B is the repn A[3,2,1] tensor A[3,2,1] t ensor A[4,2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "# which is equal to (1/n!) SUM_(perms of S_n) Chi^B(sig) * Chi^[triv] (sig^-1) \+ = = # times triv repn appears in B. " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 97 "# notice that tos(Tensored) had a 3 in front o f s[6] = trival repn appears three times in decomp." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "# We could also have computed this integer by evaluating the ex act sum" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "thesum := 0: for k from 1 to 720 d o thesum := thesum + (chilasig([3,2,1],S6[k]))*chilasig([3,2,1],S6[k]) *chilasig([4,2],S6[k]) od: thesum/6!;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "S4 2000" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "tensortest2 := tos(itensor(s [4,2],s[4,2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,tensortest2G,0&% \"sG6#\"\"'\"\"\"&F'6$\"\"&F*F**&\"\"#F*&F'6$\"\"%F/F*F*&F'6%F2F*F*F** &F/F*&F'6%\"\"$F/F*F*F*&F'6&F8F*F*F*F*&F'6%F/F/F/F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "tensortest3 := tos(itensor(tensortest2,s[ 4,2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,tensortest3G,6*&\"\"#\" \"\"&%\"sG6#\"\"'F(F(*&F,F(&F*6$\"\"&F(F(F(*&\"#8F(&F*6$\"\"%F'F(F(*& \"#5F(&F*6%F5F(F(F(F(*&F5F(&F*6$\"\"$F=F(F(*&\"#;F(&F*6%F=F'F(F(F(*&F7 F(&F*6&F=F(F(F(F(F(*&F,F(&F*6%F'F'F'F(F(*&F,F(&F*6&F'F'F(F(F(F(*&F5F(& F*6'F'F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "tenso rtest4 := tos(itensor(tensortest3,s[4,2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,tensortest4G,8*&\"#8\"\"\"&%\"sG6#\"\"'F(F(*&\"#\\F( &F*6$\"\"&F(F(F(*&\"##*F(&F*6$\"\"%\"\"#F(F(*&\"#\"*F(&F*6%F6F(F(F(F(* &\"#UF(&F*6$\"\"$F@F(F(*&\"$Y\"F(&F*6%F@F7F(F(F(*&F9F(&F*6&F@F(F(F(F(F (*&F.F(&F*6%F7F7F7F(F(*&\"#sF(&F*6&F7F7F(F(F(F(*&F=F(&F*6'F7F(F(F(F(F( F(*&F,F(&F*6(F(F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "# So A[42], A[42]*A[42], and A[42]*A[42]*A[42] (where by *, I \+ mean tensor) do not contain a copy of A[111111] (the alternating/sign \+ representation ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "# and \+ A[42]*A[42]*A[42]*A[42] contains six copies of A[111111] so the answer is four." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "S4 2002" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "# Given representations A[lambda] and A[mu] for sym metric groups S_n, S_m respectively,\n# then the outer tensor product of A[lambda] x A[mu] induces in S_(m+n) (or in S_k for k > m+n)\n# T he corresponding Schur function is s[lambda]*s[mu]. Thus in this case :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "tos(s[2,1]*s[2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.&%\"sG6$\"\"%\"\"$\"\"\"&F%6%F'\"\"#F)F)&F%6%F(F(F)F) &F%6%F(F,F,F)&F%6&F(F,F)F)F)&F%6&F,F,F,F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "# Thus the multiplicities of the different irreps of S_7 are given by the coefficients of schur function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 258 75 "S5-2002 (Note: For this problem, Denominator is supposed to be Vandermonde)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "AA:=[2,3,8, 6,1,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AAG7(\"\"#\"\"$\"\")\"\" '\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "delta:=[0,1,2, 3,4,5];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG7(\"\"!\"\"\"\"\"# \"\"$\"\"%\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Bialt:=s churf(AA+delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&BialtG,$&%\"sG6 (\"\"&F)\"\"%\"\"$F+\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# Besides the schurf command, we also can evaluate using the s linky rule: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "disp([[1,1],[2],[3,3,3,3,3,3],[4,4, 4,4,4,4,4,4],[5,5,5],[6,6]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7(7,%\"~G\"\"'F)F(F(F(F(F(F(F(7,F(\"\"&F+F+F(F(F(F(F(F(7,F(\"\" %F-F-F-F-F-F-F-F(7,F(\"\"$F/F/F/F/F/F(F(F(7,F(\"\"#F(F(F(F(F(F(F(F(7,F (\"\"\"F3F(F(F(F(F(F(F(Q*pprint5166\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "# becomes via the slinky rule" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " disp([[1,1,3,3,4],[2,3,3,4,4],[3,3,4,4],[4,4,4],[5,5,5],[6,6]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)%\"~G\"\"'F)F(F(F(F(7) F(\"\"&F+F+F(F(F(7)F(\"\"%F-F-F(F(F(7)F(\"\"$F/F-F-F(F(7)F(\"\"#F/F/F- F-F(7)F(\"\"\"F3F/F/F-F(Q*pprint5176\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "# Sign is negative since all but one (an odd number ) of the former rows now takes up an odd number of rows. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "# We can also theoreticall y use determinants directly, however this is computationally unfeasibl e because impossible to quickly tell if the two outputs are equal." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "Matri := DELTLA(Reverse(AA));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&MatriGK%'matrixG6#7(7(*$)%#x1G\"\"(\"\"\"*$)F,\"\"&F .*$)F,\"\"*F.*$)F,\"#5F.*$)F,\"\"%F.*$)F,\"\"#F.7(*$)%#x2GF-F.*$)FAF1F .*$)FAF4F.*$)FAF7F.*$)FAF:F.*$)FAF=F.7(*$)%#x3GF-F.*$)FOF1F.*$)FOF4F.* $)FOF7F.*$)FOF:F.*$)FOF=F.7(*$)%#x4GF-F.*$)FgnF1F.*$)FgnF4F.*$)FgnF7F. *$)FgnF:F.*$)FgnF=F.7(*$)%#x5GF-F.*$)FeoF1F.*$)FeoF4F.*$)FeoF7F.*$)Feo F:F.*$)FeoF=F.7(*$)%#x6GF-F.*$)FcpF1F.*$)FcpF4F.*$)FcpF7F.*$)FcpF:F.*$ )FcpF=F.Q*pprint5186\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "# DetMatri := det(Matri):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "# ExpandDet := simplify(DetMatri/Vand(6)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "# evalschur := (-1)*evalsf(s[5,5,4,3,3,2],x1+x2+x3 +x4+x5+x6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# ExpandDet = evalschur" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 259 16 "Part III R1-2002" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 " Part \+ (1)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "D6;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7.K%'matrixG6#7$7$\"\"\"\"\"!7$F*F)Q*pprint5196\"KF%6 #7$7$#F)\"\"#,$*&F3!\"\"\"\"$F2F67$,$*&F3F6F7F2F)F2Q*pprint520F-KF%6#7 $7$#F6F3F47$F9F@Q*pprint521F-KF%6#7$7$F6F*7$F*F6Q*pprint522F-KF%6#7$7$ F@F97$F4F@Q*pprint523F-KF%6#7$7$F2F97$F4F2Q*pprint524F-KF%6#7$F(FGQ*pp rint525F-KF%6#7$F1FMQ*pprint526F-KF%6#7$F?FSQ*pprint527F-KF%6#7$FFF+Q* pprint528F-KF%6#7$FLF8Q*pprint529F-KF%6#7$FRFAQ*pprint530F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "matrix(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7$#\"\"\"\"\"#,$*&F*!\"\"\"\"$F(F-7$,$ *&F*F-F.F(F)F(Q*pprint5316\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "matrix(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7$# \"\"\"\"\"#,$*&F*!\"\"\"\"$F(F)7$F+#F-F*Q*pprint5326\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# FRG is F_(R^G), i.e. the Hilbert \+ series for the ring of invariants." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "FRG := \+ hilb(D6,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FRGG,,*&\"\"\"F'*&\" #7F',(F'F'*&\"\"#F'%\"qGF'!\"\"*$)F-F,F'F'F'F.F'*&F'F'*&\"\"'F',(F/F'F -F.F'F'F'F.F'*&F'F'*&F3F',(F/F'F-F'F'F'F'F.F'*&F'F'*&F)F',(F'F'*&F,F'F -F'F'F/F'F'F.F'*&F'F'*&F,F',&F'F'F/F.F'F.F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "# By Molien's theorem, we should also be able to compute this via the sum \+ involving determinants." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "FRG2 := 0: for i to 12 \+ do FRG2:= FRG2 + 1/(det(D6[1]-q*D6[i])): od: FRG2:=(1/12)*FRG2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%FRG2G,.*&\"\"\"F'*&\"#7F'),&F'F'%\" qG!\"\"\"\"#F'F-F'*&F'F'*&\"\"'F',(*$)F,F.F'F'F,F-F'F'F'F-F'*&F'F'*&F1 F',(F3F'F,F'F'F'F'F-F'*&F'F'*&F)F'),&F,F'F'F'F.F'F-F'*&F'F'*(F1F'F+F'F ;F'F-F'*&F'F'*&\"\"$F',&F'F'F3F-F'F-F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "HILbert := factor(simplify(FRG));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(HILbertG*&\"\"\"F&**),&%\"qGF&F&!\"\"\"\"#F&),&F*F &F&F&F,F&,(*$)F*F,F&F&F*F&F&F&F&,(F0F&F*F+F&F&F&F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Pa rt (2)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "for d1 from 1 to \+ 12 do: for d2 from 1 to d1 do if (factor(1/((1-q^d1)*(1-q^d2)) - HILbe rt)=0) then print(d1, d2); fi; od; od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"'\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "# Thus \+ using d1 = 6, d2 = 2, we find indeed that the hilbert polynomial has t he form " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "1/(1-q^6)/(1-q^2);\nfactor(1/(1-q^6 )/(1-q^2));\nfactor(1/(1-q^6)/(1-q^2)-HILbert); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&,&F$F$*$)%\"qG\"\"'F$!\"\"F$,&F$F$*$)F)\"\" #F$F+F$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$**),&%\"qGF$F$! \"\"\"\"#F$),&F(F$F$F$F*F$,(*$)F(F*F$F$F(F$F$F$F$,(F.F$F(F)F$F$F$F)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "# and thus the group D6 is generated by reflections. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 8 "Part (3)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "# To calculate the first ten terms of the series, we just " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "series(HILbert,q,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"qG\"\"\"\"\"!F%\"\"#F%\"\"%F'\"\"'F'\"\")F'\"#5\"\"$\"#7F, \"#9F,\"#;F(\"#=-%\"OG6#F%\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "# Two other methods:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "G1 := 1: for i to \+ 10 do G1:=G1 + (q^2)^i: od: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "G1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8*$)%\"qG\"#5\"\"\"F(*$)F &\"\")F(F(*$)F&\"\"'F(F(*$)F&\"\"%F(F(*$)F&\"\"#F(F(F(F(*$)F&\"#7F(F(* $)F&\"#9F(F(*$)F&\"#;F(F(*$)F&\"#=F(F(*$)F&\"#?F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "GG1 := series(1/(1-q^2), q, 22); GG1 := c onvert(GG1,polynom); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GG1G+;%\"q G\"\"\"\"\"!F'\"\"#F'\"\"%F'\"\"'F'\"\")F'\"#5F'\"#7F'\"#9F'\"#;F'\"#= F'\"#?-%\"OG6#F'\"#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GG1G,8*$)% \"qG\"#5\"\"\"F**$)F(\"\")F*F**$)F(\"\"'F*F**$)F(\"\"%F*F**$)F(\"\"#F* F*F*F**$)F(\"#7F*F**$)F(\"#9F*F**$)F(\"#;F*F**$)F(\"#=F*F**$)F(\"#?F*F *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "G2:=1: for i to 10 do G2:=G2+(q^6)^i: od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "G2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8*$)%\"qG\"\"'\"\"\"F(F(F(*$)F&\"#7F(F(*$)F&\"#=F(F(*$ )F&\"#CF(F(*$)F&\"#IF(F(*$)F&\"#OF(F(*$)F&\"#UF(F(*$)F&\"#[F(F(*$)F&\" #aF(F(*$)F&\"#gF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "GG2 \+ := series(1/(1-q^6),q,66); GG2 := convert(GG2,polynom);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$GG2G+;%\"qG\"\"\"\"\"!F'\"\"'F'\"#7F'\"#=F'\" #CF'\"#IF'\"#OF'\"#UF'\"#[F'\"#aF'\"#g-%\"OG6#F'\"#m" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GG2G,8*$)%\"qG\"\"'\"\"\"F*F*F**$)F(\"#7F*F**$)F( \"#=F*F**$)F(\"#CF*F**$)F(\"#IF*F**$)F(\"#OF*F**$)F(\"#UF*F**$)F(\"#[F *F**$)F(\"#aF*F**$)F(\"#gF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "SORTED := sort(expand(G1*G2));" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>%'SORTEDG,^p*$)%\"qG\"#!)\"\"\"F**$)F(\"#yF*F**$)F(\"#wF*F**&\"\"#F *)F(\"#uF*F**&F2F*)F(\"#sF*F**&F2F*)F(\"#qF*F**&\"\"$F*)F(\"#oF*F**&F< F*)F(\"#mF*F**&F " 0 "" {MPLTEXT 1 0 29 "Ser := series(SORTED, q, 20) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SerG+9%\"qG\"\"\"\"\"!F'\"\"#F '\"\"%F)\"\"'F)\"\")F)\"#5\"\"$\"#7F.\"#9F.\"#;F*\"#=-%\"OG6#F'\"#?" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Ser := convert(Ser,polynom );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SerG,6\"\"\"F&*$)%\"qG\"\"#F& F&*$)F)\"\"%F&F&*&F*F&)F)\"\"'F&F&*&F*F&)F)\"\")F&F&*&F*F&)F)\"#5F&F&* &\"\"$F&)F)\"#7F&F&*&F8F&)F)\"#9F&F&*&F8F&)F)\"#;F&F&*&F-F&)F)\"#=F&F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 8 "Part (4)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "# The homogeneous invariants of degree 2 and 6 can be found by the Reynold's operator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Hom2 := REY(D6,x1^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Hom2G,&*&\"\"#!\"\"%#x1GF'\"\"\"*&F 'F(%#x2GF'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Hom6 := REY (D6,x1^6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Hom6G,**&#\"\"*\"#K\" \"\"*$)%#x2G\"\"'F*F*F**&#\"#6F)F**$)%#x1GF.F*F*F**&#\"#:F)F**&)F4\"\" %F*)F-\"\"#F*F*F**&#\"#XF)F**&)F4F " 0 "" {MPLTEXT 1 0 13 "REY(D6,Hom6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"*\"#K\"\"\"*$)%#x2G\"\"'F(F(F(*&#\"#6F'F(*$)%#x1GF,F(F( F(*&#\"#:F'F(*&)F2\"\"%F()F+\"\"#F(F(F(*&#\"#XF'F(*&)F2F:F()F+F8F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "# Now acting on it by a ny matrix in D6 (e.g. the generators A and B will leave it fixed." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "simplify(T(A,Hom6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"*\"#K\"\"\"*$)%#x2G\"\"'F(F(F(*&#\"#6F'F(*$)%#x1GF,F(F( F(*&#\"#:F'F(*&)F2\"\"%F()F+\"\"#F(F(F(*&#\"#XF'F(*&)F2F:F()F+F8F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(T(B,Hom6));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"*\"#K\"\"\"*$)%#x2G\"\"'F(F( F(*&#\"#6F'F(*$)%#x1GF,F(F(F(*&#\"#:F'F(*&)F2\"\"%F()F+\"\"#F(F(F(*&# \"#XF'F(*&)F2F:F()F+F8F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Part (5)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "Jac := matrix([[diff(Hom2,x1), diff (Hom2,x2)],[diff(Hom6,x1),diff(Hom6,x2)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$JacGK%'matrixG6#7$7$%#x1G%#x2G7$,(*&#\"#L\"#;\"\"\"* $)F*\"\"&F2F2F2*&#\"#:\"\")F2*&)F*\"\"$F2)F+\"\"#F2F2F2*&#\"#XF1F2*&F* F2)F+\"\"%F2F2F2,(*&#\"#FF1F2*$)F+F5F2F2F2*&#F8F1F2*&)F*FDF2F+F2F2F2*& #FAF9F2*&)F*F>F2)F+F " 0 " " {MPLTEXT 1 0 9 "det(Jac);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\" \"*\"\")\"\"\"*&%#x1GF()%#x2G\"\"&F(F(!\"\"*&#F&F'F(*&)F*F-F(F,F(F(F.* &#\"#:\"\"%F(*&)F*\"\"$F()F,F9F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "# Also could have used jacobian command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "jacobian([Hom2,Hom6],[x1,x2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7$%#x1G%#x2G7$,(*&#\"#L\"#;\"\"\"*$)F(\" \"&F0F0F0*&#\"#:\"\")F0*&)F(\"\"$F0)F)\"\"#F0F0F0*&#\"#XF/F0*&F(F0)F) \"\"%F0F0F0,(*&#\"#FF/F0*$)F)F3F0F0F0*&#F6F/F0*&)F(FBF0F)F0F0F0*&#F?F7 F0*&)F(F " 0 "" {MPLTEXT 1 0 103 "# Since the Jacobian is nontrivial, this means that \+ these two invariants are algebraically independent." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "testAI(Hom6,[Hom2],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)% #x2G\"\"'\"\"\"!\"\"*(\"\"$F()F&\"\"%F(%#y1GF(F(*&#\"\"*F-F(*&)F&\"\"# F()F.F4F(F(F)*&#\"#6F-F(*$)F.F+F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "# The fact that this expression involves x's in addi tion to y's also implies the two invariants are algebraically independ ent" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 5 "Hom6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**& #\"\"*\"#K\"\"\"*$)%#x2G\"\"'F(F(F(*&#\"#6F'F(*$)%#x1GF,F(F(F(*&#\"#:F 'F(*&)F2\"\"%F()F+\"\"#F(F(F(*&#\"#XF'F(*&)F2F:F()F+F8F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Hom2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#!\"\"%#x1GF%\"\"\"*&F%F&%#x2GF%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Part (6)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Hom2;\nHom 6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#!\"\"%#x1GF%\"\"\"*&F%F &%#x2GF%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"*\"#K\"\"\"*$)% #x2G\"\"'F(F(F(*&#\"#6F'F(*$)%#x1GF,F(F(F(*&#\"#:F'F(*&)F2\"\"%F()F+\" \"#F(F(F(*&#\"#XF'F(*&)F2F:F()F+F8F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Jac := jacobian([Hom2,Hom6],[x1,x2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$JacGK%'matrixG6#7$7$%#x1G%#x2G7$,(*&#\"#L\"#;\" \"\"*$)F*\"\"&F2F2F2*&#\"#:\"\")F2*&)F*\"\"$F2)F+\"\"#F2F2F2*&#\"#XF1F 2*&F*F2)F+\"\"%F2F2F2,(*&#\"#FF1F2*$)F+F5F2F2F2*&#F8F1F2*&)F*FDF2F+F2F 2F2*&#FAF9F2*&)F*F>F2)F+F \+ " 0 "" {MPLTEXT 1 0 9 "det(Jac);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,( *&#\"\"*\"\")\"\"\"*&%#x1GF()%#x2G\"\"&F(F(!\"\"*&#F&F'F(*&)F*F-F(F,F( F(F.*&#\"#:\"\"%F(*&)F*\"\"$F()F,F9F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "factor(det(Jac),sqrt(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*0\"\")!\"\",&*&\"\"$\"\"\"%#x2GF*F**&%#x1GF*F)#F*\" \"#F*F*,&*&F)F*F+F*F*F,F&F*,&F+F*F,F*F*,&F+F*F,F&F*F+F*F-F*F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# so the six lines are:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "# x1 = 0" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "# x2 = 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "# (sqrt(3)*x1 = x2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "# (sqrt(3)*x1 = -x2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "# x1 = (sqrt(3)*x2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "# x1 = -(sqrt(3)*x2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "# The reflections of the (x1,x2)-pl ane about these six lines generate D6." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Part (7)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "GB:=gbasis([Hom2-y1,Hom6-y2] ,[x1,x2,y1,y2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GBG7$,,*&\"\"% \"\"\")%#x2G\"\"'F)F)*(\"#7F))F+F(F)%#y1GF)!\"\"*(\"\"*F))F+\"\"#F))F0 F5F)F)*&\"#6F))F0\"\"$F)F1*&F(F)%#y2GF)F),(*$)%#x1GF5F)F)*$F4F)F)*&F5F )F0F)F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "map(print,GB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"\"%\"\"\")%#x2G\"\"'F&F&*(\"#7 F&)F(F%F&%#y1GF&!\"\"*(\"\"*F&)F(\"\"#F&)F-F2F&F&*&\"#6F&)F-\"\"$F&F.* &F%F&%#y2GF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%#x1G\"\"#\"\" \"F(*$)%#x2GF'F(F(*&F'F(%#y1GF(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "# Thus gbasis for \+ is: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "eval(GB,[y1=0,y2=0]); # y1 a nd y2 are in the ideal and hence equivalent to 0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*&\"\"%\"\"\")%#x2G\"\"'F'F',&*$)%#x1G\"\"#F'F'*$)F )F/F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# Notice since s et them to zero anyway, could have:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gbasis( [Hom2,Hom6],[x1,x2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$)%#x2G\" \"'\"\"\",&*$)%#x1G\"\"#F(F(*$)F&F-F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Part (8)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "# A monomial basis for the \+ quotient ring is \{1, x2, x2^2, x2^3, x2^4, x2^5, x1, x1*x2, x1*x2^2 , x1*x2^3, x1*x2^4, x1*x2^5\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "# Q[x1,x2]/ by our theorems is finite dimensional; \+ also, since D6 is generated by reflections, its" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 98 "# dimension is exactly |D6| = 12. Further, thi s basis is given by every monomial of smaller degree" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# than the elements in the Grobner basis. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 76 "eval(GB,[y1=0,y2=0]); # y1 and y2 are in the i deal and hence equivalent to 0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$ *&\"\"%\"\"\")%#x2G\"\"'F'F',&*$)%#x1G\"\"#F'F'*$)F)F/F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "FIND := findbas(GB, 2, 10 ); nops(F IND);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%FINDG7.\"\"\"%#x1G%#x2G*&F 'F&F(F&*$)F(\"\"#F&*&F'F&F+F&*$)F(\"\"$F&*&F'F&F/F&*$)F(\"\"%F&*&F'F&F 3F&*$)F(\"\"&F&*&F'F&F7F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "# findbas finds a monomial \+ basis for the quotient by running through all the monomials in 2 vars \+ up to degree 10 and modding by GB " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "# I used 10 arbitrarily because wanted to use a degr ee high enough so it is definitely killed by GB " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "# Any other monomial in the quotient is eith er zero or can be rewritten in terms of the smaller monomial in above \+ basis. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Part (9)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# Hilbert series = Sum over i from 0 to 6 of number m onomials of degree i" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Rstarhilb := (1+q)*(1+q+q^2 +q^3+q^4 + q^5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*RstarhilbG*&,&% \"qG\"\"\"F(F(F(,.F(F(F'F(*$)F'\"\"#F(F(*$)F'\"\"$F(F(*$)F'\"\"%F(F(*$ )F'\"\"&F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sort(expa nd(Rstarhilb));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$)%\"qG\"\"'\"\" \"F(*&\"\"#F()F&\"\"&F(F(*&F*F()F&\"\"%F(F(*&F*F()F&\"\"$F(F(*&F*F()F& F*F(F(*&F*F(F&F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Part (10) and (11)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "for i from 1 to 6 do; Allinvar(D6,i ,2); od; # All lin ind invariants of D6 with deg <= 6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#,&*&\"\"# !\"\"%#x1GF&\"\"\"*&F&F'%#x2GF&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#< \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#,(*&#\"\"$\"\")\"\"\"*$)%#x1G \"\"%F)F)F)*&#F'F-F)*&)F,\"\"#F))%#x2GF2F)F)F)*&F&F)*$)F4F-F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<$,**&#\"\"*\"#K\"\"\"*$)%#x2G\"\"'F)F)F)*&#\"#6F(F)*$)%#x1GF-F)F)F)* &#\"#:F(F)*&)F3\"\"%F))F,\"\"#F)F)F)*&#\"#XF(F)*&)F3F;F))F,F9F)F)F),** &#\"\"$F(F)F*F)F)*&#F)F(F)F1F)F)*&#\"#@F(F)F7F)F)*&#F'F(F)F?F)!\"\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "# Any degree 6 invariant will be a linear combina otion of the two in the last line. Thus we apply the # Reynold's oper ator to an arbitrary polynomial of degree 6." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "a0 *x1^6+a1*x1^5*x2+a2*x1^4*x2^2+a3*x1^3*x2^3+a4*x1^2*x2^4+a5*x1*x2^5+a6* x2^6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&%#a0G\"\"\")%#x1G\"\"'F&F &*(%#a1GF&)F(\"\"&F&%#x2GF&F&*(%#a2GF&)F(\"\"%F&)F.\"\"#F&F&*(%#a3GF&) F(\"\"$F&)F.F8F&F&*(%#a4GF&)F(F4F&)F.F2F&F&*(%#a5GF&F(F&)F.F-F&F&*&%#a 6GF&)F.F)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Gen6Invar \+ := REY(D6,a0*x1^6+a1*x1^5*x2+a2*x1^4*x2^2+a3*x1^3*x2^3+a4*x1^2*x2^4+a5 *x1*x2^5+a6*x2^6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Gen6InvarG,B* ,\"#:\"\"\"\"#K!\"\"%#a0GF(%#x1G\"\"%%#x2G\"\"#F(*(F)F*%#a2GF(F,\"\"'F (**\"\"$F(F)F*F1F(F.F2F(*,\"#XF(F)F*F+F(F,F/F.F-F(*,F6F(F)F*%#a6GF(F,F -F.F/F(*,F'F(F)F*F8F(F,F/F.F-F(*,\"\"*F(F)F*F1F(F,F/F.F-F***F4F(F)F*%# a4GF(F,F2F(*(F)F*F=F(F.F2F(**\"#6F(F)F*F+F(F,F2F(**F@F(F)F*F8F(F.F2F(* ,F;F(F)F*F=F(F,F-F.F/F**,\"#@F(F)F*F1F(F,F-F.F/F(*,FDF(F)F*F=F(F,F/F.F -F(**F;F(F)F*F+F(F.F2F(**F;F(F)F*F8F(F,F2F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# Could also write general 6th degree invariant as :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 131 "Oth := expand(c1*(11/32*x2^6+15/32*x2^4*x1^2+ 45/32*x2^2*x1^4+9/32*x1^6) + c2*(1/32*x2^6+21/32*x2^4*x1^2-9/32*x2^2*x 1^4+3/32*x1^6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$OthG,2**\"#6\" \"\"\"#K!\"\"%#c1GF(%#x2G\"\"'F(*,\"#:F(F)F*F+F(%#x1G\"\"#F,\"\"%F(*, \"#XF(F)F*F+F(F0F2F,F1F(**\"\"*F(F)F*F+F(F0F-F(*(F)F*%#c2GF(F,F-F(*,\" #@F(F)F*F8F(F0F1F,F2F(*,F6F(F)F*F8F(F0F2F,F1F***\"\"$F(F)F*F8F(F0F-F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "Cubed := expand(Hom2^3 ); # Cubing a second degree invariant yields a 6th degree invariant t hat is # indeed a linear combination of the sixth degree invariants." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&CubedG,**&#\"\"\"\"\")F(*$)%#x1G \"\"'F(F(F(*&#\"\"$F)F(*&)F,\"\"%F()%#x2G\"\"#F(F(F(*&F/F(*&)F,F6F()F5 F3F(F(F(*&F'F(*$)F5F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "testAI(Cubed,[Inv6[ 1],Inv6[2]],2); # Since dimension of space of 6th degree invariants \+ is two, we only use the first two as our basis for that space" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\")F&*$)%#x1G\"\"'F&F&F&* &#\"\"$F'F&*&)F*\"\"%F&)%#x2G\"\"#F&F&F&*&F-F&*&)F*F4F&)F3F1F&F&F&*&F% F&*$)F3F+F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "# So (Hom2^3) is 1/3 of the \+ sum of the first two 6th degree homogeneous invariants." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# Writing the other invariants in t erms of Hom2 and Hom6. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "testAI(Allinvar(D6,6,2) [1],[Hom2,Hom6],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#y2G" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "testAI(Allinvar(D6,6,2)[2],[ Hom2,Hom6],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"$\"\"\")%#y1 GF%F&F&%#y2G!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "Rand6 Inv := eval(Gen6Invar,[a0=1,a1=-1,a2=5,a3=3,a4=-2,a5=0,a6=0]); # A ra ndom 6th degree invariant" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Rand6I nvG,**&#\"#p\"#;\"\"\"*&)%#x1G\"\"%F*)%#x2G\"\"#F*F*F**&#\"\"&F)F**$)F -\"\"'F*F*F**&#\"#6F)F**$)F0F7F*F*F**&#\"#@F)F**&)F-F1F*)F0F.F*F*!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "testAI(Rand6Inv,[Hom2,Hom6],2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&\"#>\"\"\")%#y1G\"\"$F&F&*&\"\"'F&%#y2GF&!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "# We can also find t he polynomial that writes a random 6th degree invariant in terms of Ho m2 and Hom6 (note that Hom2^3 = Cubed as above)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "testAI(Gen6Invar,[Hom2,Hom6],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &*&,*%#a2G!\"\"%#a0G\"\"\"%#a4GF)%#a6GF'F)%#y2GF)F)*&,(*&\"\"$F)F&F)F) *&\"\"#F)F*F)F'*&\"\"&F)F+F)F)F))%#y1GF0F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "testAI(Oth,[Hom2,Hom6],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&%#c1G!\"\"%#c2G\"\"\"F)%#y2GF)F)*&,&*&\"\"&F)F&F) F)*&\"\"#F)F(F)F'F))%#y1G\"\"$F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "# Moral: \{Hom2^3, Hom 6\} yield a basis for the spa ce of 6th degree invariants, and since there is only 2nd degree invari ant, \{Hom2\} is a basis for that space " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Allin var(D6,4,2); # The 4th degree invariants are" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#,(*&#\"\"$\"\")\"\"\"*$)%#x1G\"\"%F)F)F)*&#F'F-F)*&)F ,\"\"#F))%#x2GF2F)F)F)*&F&F)*$)F4F-F)F)F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "testA I(Allinvar(D6,4,2)[1],[Hom2],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *(\"\"$\"\"\"\"\"#!\"\"%#y1GF'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "JJ := jacobian([Allinvar2(D6,4)[1],Hom2],[x1,x2]); \+ # Computing Jacobian also will verify Algebraic Dependence" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#JJGK%'matrixG6#7$7$,&*&#\"\"$\"\"#\"\"\"* $)%#x1GF-F/F/F/*&F,F/*&F2F/)%#x2GF.F/F/F/,&*&F,F/*&)F2F.F/F6F/F/F/*&F, F/*$)F6F-F/F/F/7$F2F6Q*pprint5366\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "det(JJ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 290 "# so \{Hom2^2\} is also a basis for the space of 4 th degree invariants and so in total, \{Hom2, Hom6\} generate all of t he invariants of degree 6 or less\n# and as the denominator of the Hil bert Series tells us, this is sufficient to say that \{Hom2, Hom6\} ge nerate the entire Ring of Invariants." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 256 "" 0 "" {TEXT -1 19 "Part III R.2.1-2002" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Part (1)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "A4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.7&\"\"\"\"\"# \"\"$\"\"%7&F%F'F(F&7&F%F(F&F'7&F&F%F(F'7&F&F'F%F(7&F&F(F'F%7&F'F%F&F( 7&F'F&F(F%7&F'F(F%F&7&F(F%F'F&7&F(F&F%F'7&F(F'F&F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "prep([2,1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(Q*pprint53 76\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "NG := map(NAT,A4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NGG7.K%'matrixG6#7%7%\"\"\"\"\" !F,7%F,F+F,7%F,F,F+Q*pprint5386\"KF'6#7%7%F,!\"\"F,7%F,F,F5F*Q*pprint5 39F0KF'6#7%F.7%F5F,F,F4Q*pprint540F0KF'6#7%7%F,F5F57%F5F,F+F6Q*pprint5 41F0KF'6#7%7%F+F+F,F@F-Q*pprint542F0KF'6#7%FAFFF;Q*pprint543F0KF'6#7%7 %F+F,F5F.F@Q*pprint544F0KF'6#7%7%F5F5F,F*FAQ*pprint545F0KF'6#7%7%F,F+F +F4FFQ*pprint546F0KF'6#7%F6FYFOQ*pprint547F0KF'6#7%F-FTFYQ*pprint548F0 KF'6#7%F;FOFTQ*pprint549F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "HiL := hilb(NG,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$HiLG,(*& \"\"\"F'*&\"#7F',*F'F'*&\"\"$F'%\"qGF'!\"\"*&F,F')F-\"\"#F'F'*$)F-F,F' F.F'F.F'*(F1F'F,F.,&F'F'F2F.F.F'*&F'F'*&\"\"%F',*F'F'F-F'*$F0F'F.F2F.F 'F.F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(HiL);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,(*$)%\"qG\"\"%\"\"\"F**$)F(\"\"#F *!\"\"F*F*F*,*F*F.F(F.F+F**$)F(\"\"$F*F*F.,&F0F*F*F.F.,&F(F*F*F.F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Part (2)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "for d1 from 1 to 4 do; for d2 from 1 to d1 do; for d3 from 1 to d 2 do; for d4 from 1 to 12 do; if simplify( (1+q^d4)/ ((1-q^d1)*(1-q^d2 )*(1-q^d3)) - HiL) = 0\nthen print([d1,d2,d3,d4]); fi; od; od; od; od; # Would really be loop from 1 to 12 if not precomputed" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7&\"\"%\"\"$\"\"#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "HILS := (1+q^6)/ ((1-q^4)*(1-q^3)*(1-q^2));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "simplify(HILS-HiL);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%HILSG**,&\"\"\"F'*$)%\"qG\"\"'F'F'F',&F'F'*$)F* \"\"%F'!\"\"F0,&F'F'*$)F*\"\"$F'F0F0,&F'F'*$)F*\"\"#F'F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Part (3)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "series(HILS,q,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"qG\"\"\"\"\"!F%\"\"#F%\"\"$F'\"\"%F%\"\"&F)\" \"'F'\"\"(F*\"\")F)\"\"*F,\"#5-%\"OG6#F%\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Part (4 ) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "# The fact that the n umerator has terms 1 and q^d4 means that the separators of the system \+ are \{1,eta\} where eta is an invariant of degree d4 = 6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "# The fact that the denominator ha s the form (1-q^d1)*(1-q^d2)*(1-q^d3) means that there are three quasi -generators of degrees d1=2, d2=3, d3=4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 10 "Part (5-6) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "for i from 1 to 4 do; I nvar[i] := Allinvar(NG,i,3); od; # All the homogeneous invariants of o rder 4 or less (I used Allivar since faster for these degrees)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&InvarG6#\"\"\"<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&InvarG6#\"\"#<#,.*&#\"\"\"F'F,*$)%#x1GF'F,F,F, *&F+F,*$)%#x2GF'F,F,F,*&F+F,*$)%#x3GF'F,F,F,*&#F,F'F,*&F/F,F3F,F,!\"\" *&F+F,*&F/F,F7F,F,F,*&#F,F'F,*&F3F,F7F,F,F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&InvarG6#\"\"$<#,0*&#F'\"\"%\"\"\"*&)%#x1G\"\"#F-%#x 2GF-F-F-*&#F'F,F-*&F/F-%#x3GF-F-!\"\"*&#F'F,F-*&F0F-)F2F1F-F-F7*&#F'F1 F-*(F0F-F2F-F6F-F-F-*&#F'F,F-*&F0F-)F6F1F-F-F7*&#F'F,F-*&F;F-F6F-F-F7* &F+F-*&F2F-FBF-F-F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%&InvarG6#\" \"%<$,@*&#\"\"\"\"\"#F,*$)%#x3GF'F,F,F,*&F+F,*$)%#x2GF'F,F,F,*&F+F,*$) %#x1GF'F,F,F,*&#\"\"$F-F,*&)F8F-F,)F4F-F,F,F,*&F8F,)F4F;F,!\"\"*&)F8F; F,F4F,FA*&F:F,*&F=F,)F0F-F,F,F,**F;F,F=F,F4F,F0F,FA**F;F,F8F,F>F,F0F,F ,**F;F,F8F,F4F,FFF,FA*&FCF,F0F,F,*&F8F,)F0F;F,F,*&F@F,F0F,FA*&F4F,FLF, FA*&F:F,*&F>F,FFF,F,F,,@*&F+F,FF,F0F,F,F,*&#F,F;F,*(F8F,F4F,FFF,F,FA*&F[oF,FJ F,F,*&F[oF,FKF,F,*&#F,F;F,FMF,FA*&#F,F;F,FNF,FA*&F+F,FPF,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "I2 := Invar[2][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I2G,.*&#\"\"\"\"\"#F(*$)%#x1GF)F(F(F(*&F' F(*$)%#x2GF)F(F(F(*&F'F(*$)%#x3GF)F(F(F(*&#F(F)F(*&F,F(F0F(F(!\"\"*&F' F(*&F,F(F4F(F(F(*&#F(F)F(*&F0F(F4F(F(F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "I3 := Invar[3][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#I3G,0*&#\"\"$\"\"%\"\"\"*&)%#x1G\"\"#F*%#x2GF*F*F**&#F(F)F**&F,F*% #x3GF*F*!\"\"*&#F(F)F**&F-F*)F/F.F*F*F4*&#F(F.F**(F-F*F/F*F3F*F*F**&#F (F)F**&F-F*)F3F.F*F*F4*&#F(F)F**&F8F*F3F*F*F4*&F'F**&F/F*F?F*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "I4 := Invar[4][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I4G,@*&#\"\"\"\"\"#F(*$)%#x3G\"\"%F(F(F(* &F'F(*$)%#x2GF-F(F(F(*&F'F(*$)%#x1GF-F(F(F(*&#\"\"$F)F(*&)F5F)F()F1F)F (F(F(*&F5F()F1F8F(!\"\"*&)F5F8F(F1F(F>*&F7F(*&F:F()F,F)F(F(F(**F8F(F:F (F1F(F,F(F>**F8F(F5F(F;F(F,F(F(**F8F(F5F(F1F(FCF(F>*&F@F(F,F(F(*&F5F() F,F8F(F(*&F=F(F,F(F>*&F1F(FIF(F>*&F7F(*&F;F(FCF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "J := jacobian([I2,I3,I4],[x1,x2,x3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JGK%'matrixG6#7%7%,(%#x1G\"\"\"*& \"\"#!\"\"%#x2GF,F/*&F.F/%#x3GF,F,,(F0F,*&F.F/F+F,F/*&F.F/F2F,F/,(F2F, *&F.F/F+F,F,*&F.F/F0F,F/7%,,*&#\"\"$F.F,*&F+F,F0F,F,F,*&#F=F.F,*&F+F,F 2F,F,F/*&#F=\"\"%F,*$)F0F.F,F,F/*&FF,F/*&FF,F,*&#F=F.F,FAF,F/*&#F=FDF,FEF,F /*&F " 0 "" {MPLTEXT 1 0 15 "factor(det(J));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*2\"\"$\"\"\"\"\")!\"\",&%#x2GF&%#x1GF&F&,&%#x3GF&F +F(F&,&F-F&F*F&F&,(F*F(*&\"\"#F&F-F&F&F+F&F&,(F*F(F-F&*&F1F&F+F&F&F&,( *&F1F&F*F&F(F-F&F+F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "# jacobian nonzero so algebraically independent, also:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "testAI(I2,[I3,I4],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&#\"\"\"\"\"#F&*$)%#x1GF'F&F&F&*&F%F&*$)%#x2GF'F&F&F&*&F%F&*$ )%#x3GF'F&F&F&*&#F&F'F&*&F*F&F.F&F&!\"\"*&F%F&*&F*F&F2F&F&F&*&#F&F'F&* &F.F&F2F&F&F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "testAI(I3, [I2,I4],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&#\"\"$\"\"%\"\"\"*$ )%#x2GF&F(F(!\"\"*&#F&F'F(*&)F+\"\"#F(%#x3GF(F(F(*&#F&F'F(*&F+F()F2F1F (F(F,*&F.F(*$)F2F&F(F(F(*&#F&F1F(*&F+F(%#y1GF(F(F(*&#F&F1F(*&F2F(F=F(F (F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "testAI(I4,[I2,I3],3) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%#x3G\"\"%\"\"\"F(*(\"\"#F() F&F*F(%#y1GF(!\"\"*&F*F()F,F*F(F(*&#F'\"\"$F(*&%#y2GF(F&F(F(F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 13 "Part (6 b -7)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "INV6 := Allinvar3(NG,6); nops(INV6); # The homogeneous invariants of degree 6\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%INV6G<),B*&#\"\" \"\"\"%F)*&)%#x2GF*F))%#x3G\"\"#F)F)F)*&#F)F0F)*&)F-\"\"$F))F/F5F)F)! \"\"*&F(F)*&)F-F0F))F/F*F)F)F)*&#F)F0F)*&)%#x1GF5F)F4F)F)F7*&#F)F0F)*( )F@F0F)F-F)F6F)F)F7*(FDF)F:F)F.F)F)*&#F)F0F)*(FDF)F4F)F/F)F)F7*&#F)F0F )*(F?F)F:F)F/F)F)F)*&#F)F0F)*(F?F)F-F)F.F)F)F7*&#F)F0F)*(F@F)F4F)F.F)F )F7*&FJF)*(F@F)F:F)F6F)F)F)*&F(F)*&FDF)F;F)F)F)*&FJF)*&F?F)F6F)F)F)*&F (F)*&)F@F*F)F.F)F)F)*&F(F)*&FZF)F:F)F)F)*&F(F)*&FDF)F,F)F)F),Z*&#F5F*F )F+F)F)*&#F0F5F)F3F)F7*&#F)F0F)*&)F-\"\"&F)F/F)F)F7*&F[oF)F9F)F)*&#F)F 0F)*&F-F))F/FboF)F)F7*&#F)F0F)*&F@F)FaoF)F)F7*&#F)F0F)*&)F@FboF)F-F)F) F7*&#F0F5F)F>F)F7*&#\"\"(\"\"'F)*(F@F)F,F)F/F)F)F)*&#F*F5F)FCF)F7*&#Fc pFdpF)*(FZF)F-F)F/F)F)F7*&#F5F0F)FEF)F)*&#F*F5F)FHF)F7*&#F*F5F)FKF)F)* &#F*F5F)FNF)F7*&#FcpFdpF)*(F@F)F-F)F;F)F)F7*&#F*F5F)FQF)F7*&F`qF)FSF)F )*&F[oF)FUF)F)*&FJF)*&F^pF)F/F)F)F)*&FJF)*&F@F)FgoF)F)F)*&#F0F5F)FWF)F )*&F[oF)FYF)F)*&#F)FdpF)*$)F-FdpF)F)F)*&FbrF)*$)F@FdpF)F)F)*&F[oF)FfnF )F)*&F[oF)FhnF)F)*&FbrF)*$)F/FdpF)F)F),Z*&#\"#:F*F)F+F)F)*(FboF)F4F)F6 F)F7*&#F5F0F)F`oF)F7*&F_sF)F9F)F)*&#F5F0F)FfoF)F7*&#F5F0F)FjoF)F7*&#F5 F0F)F]pF)F7*(FboF)F?F)F4F)F7*&#F`sF0F)FepF)F)**F`sF)FDF)F-F)F6F)F7*&#F `sF0F)FjpF)F7*&#\"#XF0F)FEF)F)**F`sF)FDF)F4F)F/F)F7**F`sF)F?F)F:F)F/F) F)**F`sF)F?F)F-F)F.F)F7*&#F`sF0F)FeqF)F7**F`sF)F@F)F4F)F.F)F7**F`sF)F@ F)F:F)F6F)F)*&F_sF)FUF)F)*&F\\qF)F[rF)F)*&F\\qF)F]rF)F)*(FboF)F?F)F6F) F)*&F_sF)FYF)F)*&FJF)FcrF)F)*&FJF)FfrF)F)*&F_sF)FfnF)F)*&F_sF)FhnF)F)* &FJF)F[sF)F),T*&#F)F5F)F+F)F)*&#F)F*F)F3F)F7*&#F)\"#7F)F`oF)F7*&#F)F\\ vF)F9F)F7*&#F)F\\vF)FfoF)F)*&F`vF)FjoF)F)*&#F)F\\vF)F]pF)F7*&#F)F*F)F> F)F7*&F`vF)FepF)F)*&#FcpF\\vF)FCF)F7*&#F)F\\vF)FjpF)F7*&F[oF)FEF)F)*&# F)F*F)FHF)F7*&#FcpF\\vF)FKF)F)*&#F)F*F)FNF)F7*&#F)F\\vF)FeqF)F7*&#FcpF \\vF)FQF)F7*&F(F)FSF)F)*&FguF)FUF)F)*&#F)F\\vF)F[rF)F7*&F`vF)F]rF)F)*& F(F)FWF)F)*&#F)F\\vF)FYF)F7*&FguF)FfnF)F)*&#F)F\\vF)FhnF)F7,H*&#F)F*F) F+F)F7*&FJF)F3F)F)*&#F)F*F)F9F)F7*&FJF)F>F)F)*&#F0F5F)FepF)F7*&#\"#6Fd pF)FCF)F)*&F_rF)FjpF)F)**F5F)FDF)F:F)F.F)F7*&F[yF)FHF)F)*&#F\\yFdpF)FK F)F7*&F[yF)FNF)F)*&F_rF)FeqF)F)*&F[yF)FQF)F)*&#F\\yFdpF)FSF)F7*&#F)F*F )FUF)F7*&#F)F0F)FWF)F7*&#F)F*F)FYF)F7*&#F)F*F)FfnF)F7*&#F)F*F)FhnF)F7, T*&#F)F\\vF)F+F)F7*&#F)F*F)F3F)F7*&F`vF)F`oF)F)*&FguF)F9F)F)*&#F)F\\vF )FfoF)F7*&#F)F\\vF)FjoF)F7*&F`vF)F]pF)F)*&#F)F*F)F>F)F7*&F`vF)FepF)F)* &#F)F*F)FCF)F7*&#F)F\\vF)FjpF)F7*&F[oF)FEF)F)*&#FcpF\\vF)FHF)F7*&F(F)F KF)F)*&#FcpF\\vF)FNF)F7*&#F)F\\vF)FeqF)F7*&#F)F*F)FQF)F7*&F_wF)FSF)F)* &#F)F\\vF)FUF)F7*&F`vF)F[rF)F)*&#F)F\\vF)F]rF)F7*&F(F)FWF)F)*&FguF)FYF )F)*&#F)F\\vF)FfnF)F7*&FguF)FhnF)F),X*&#F)F0F)F+F)F7*&FbrF)F3F)F)*&FJF )F`oF)F)*&#F)F0F)F9F)F7*&FJF)FfoF)F)*&FJF)FjoF)F)*&FJF)F]pF)F)*&FbrF)F >F)F)FepF7*&FJF)FCF)F)FjpF)*&FJF)FHF)F)*&#F)F0F)FKF)F7*&FJF)FNF)F)FeqF )*&FJF)FQF)F)*&#F)F0F)FSF)F7*&#F)F0F)FUF)F7*&#F)F0F)F[rF)F7*&#F)F0F)F] rF)F7*&#F)FdpF)FWF)F7*&#F)F0F)FYF)F7*&#F)FdpF)FcrF)F7*&#F)FdpF)FfrF)F7 *&#F)F0F)FfnF)F7*&#F)F0F)FhnF)F7*&#F)FdpF)F[sF)F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for i from 1 to nops(INV6) do testAI(INV6[i],[I2,I3,I4],3); od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "# So two invariants outside ideal but other five inside" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%#y 1G\"\"\"%#y3GF&!\"\"*&\"\"#F&)F%\"\"$F&F&*&#\"\"%\"\"*F&*$)%#y2GF*F&F& F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"$F&*&%#y1GF&%#y3G F&F&F&*&#\"\"#F'F&*$)F)F'F&F&F&*&#\"\"%\"\"*F&*$)%#y2GF-F&F&!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"$\"\"\"%#y1GF&%#y3GF&F&*&\"\"# F&)F'F%F&!\"\"*&#\"\"%F%F&*$)%#y2GF*F&F&F&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,T*(%#x1G\"\"\")%#x2G\"\"#F&)%#x3G\"\"$F&!\"\"*,F)F&F,F -F(F&F+F,%#y1GF&F&**F,F-F%F&F+F,F/F&F&*(F,F-F+F,%#y2GF&F-*,F)F&F,F-F(F &F/F)F+F&F-*,F)F&\"\"*F-F(F&F/F&F2F&F-**F,F-F%F&F+F)F2F&F-*,\"\"%F&F5F -F+F&F/F&F2F&F&**F)F&F,F-F+F)F/F)F-*(F,F-F(F)%#y3GF&F-*(F8F&F,F-F/F,F& *,F8F&F,F-F%F&F(F&F/F)F-**F%F&F'F&F+F&F/F&F&**F)F&F,F-F/F&F;F&F-*(F,F- F+F)F;F&F&**F,F-F%F&F(F)F2F&F&*.F)F&F,F-F%F&F(F&F+F)F/F&F&**F,F-F(F)F+ F)F/F&F-**F,F-F(F)F+F&F2F&F-*.F)F&F,F-F%F&F(F&F+F&F2F&F&**F)F&F,F-F(F) F/F)F&*(F8F&F5F-F2F)F&*,F)F&F5F-F/F&F%F&F2F&F-*,F)F&F,F-F%F&F(F&F;F&F& **F,F-F%F&F+F&F;F&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\" \"$F&*&%#y1GF&%#y3GF&F&!\"\"*&#\"\"#F'F&*$)F)F'F&F&F&*&#\"\"%\"\"*F&*$ )%#y2GF.F&F&F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,N*(%#x1G\"\"\")%#x2 G\"\"#F&)%#x3G\"\"$F&F&**F%F&F'F&F+F&%#y1GF&!\"\"*.F)F&F,F/F%F&F(F&F+F )F.F&F/**F,F/F(F)F+F)F.F&F&**F,F/F%F&F+F,F.F&F/*,F)F&F,F/F(F&F+F,F.F&F /*,\"\"%F&F,F/F%F&F(F&F.F)F&**F)F&F,F/F(F)F.F)F/*,F)F&F,F/F(F&F.F)F+F& F&**F)F&F,F/F+F)F.F)F&**F,F/F%F&F(F)%#y2GF&F/*.F)F&F,F/F%F&F(F&F+F&F:F &F/**F,F/F(F)F+F&F:F&F&**F,F/F%F&F+F)F:F&F&*(F,F/F+F,F:F&F&*,F)F&\"\"* F/F.F&F%F&F:F&F&*,F)F&F@F/F(F&F.F&F:F&F&*,F5F&F@F/F+F&F.F&F:F&F/*,F)F& F,F/F%F&F(F&%#y3GF&F/*(F,F/F(F)FDF&F&**F,F/F%F&F+F&FDF&F&*(F,F/F+F)FDF &F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%#y1G\"\"\"%#y3GF&!\"\"*&# \"\"#\"\"$F&*$)F%F,F&F&F&*&#\"\")\"\"*F&*$)%#y2GF+F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "# Due to the way Maple is programm ed, it outputs these in different order each time so I manually enter \+ eta and eta2 as the two independent invariants" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "unassign('eta','eta2');" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 386 "eta := 7/12*x2^2*x3^3*x1-1/4*x2^3*x3^2*x1-1/4 *x2*x3^3*x1^2-7/12*x2^3*x1^2*x3-7/12*x2*x3^2*x1^3+1/4*x2^2*x1^3*x3-1/1 2*x2*x3^5+1/3*x1^4*x3^2+1/12*x1^5*x3-1/12*x1*x3^5+1/3*x2^4*x1^2-1/12*x 2^2*x1^4+1/12*x1^5*x2+1/12*x3*x2^5-1/12*x3^2*x2^4-1/12*x1*x2^5-1/4*x3^ 3*x2^3-1/12*x3^4*x1^2-1/12*x3^4*x1*x2+1/12*x1*x3*x2^4+1/4*x1^3*x3^3+3/ 4*x1^2*x2^2*x3^2-1/12*x1^4*x2*x3-1/4*x1^3*x2^3+1/3*x3^4*x2^2:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 387 "eta2 := 1/4*x2^2*x3^3*x1-7/ 12*x2^3*x3^2*x1-7/12*x2*x3^3*x1^2-1/4*x2^3*x1^2*x3-1/4*x2*x3^2*x1^3+7/ 12*x2^2*x1^3*x3+1/12*x2*x3^5-1/12*x1^4*x3^2-1/12*x1^5*x3+1/12*x1*x3^5- 1/12*x2^4*x1^2+1/3*x2^2*x1^4-1/12*x1^5*x2-1/12*x3*x2^5+1/3*x3^2*x2^4+1 /12*x1*x2^5-1/4*x3^3*x2^3+1/3*x3^4*x1^2-1/12*x3^4*x1*x2+1/12*x1*x3*x2^ 4+1/4*x1^3*x3^3+3/4*x1^2*x2^2*x3^2-1/12*x1^4*x2*x3-1/4*x1^3*x2^3-1/12* x3^4*x2^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "# To find th e polynomial P such that P(I2,I3,I4,eta) = 0, I use the SepPoly comman d that I programmed above in the Invariant Theory Procs section" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "Sep := SepPoly(eta,[I2,I3,I4],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SepG<#,8*&\"#O\"\"\")%#y1G\"\"'F)F)*(\"$g\"F))%#y 2G\"\"#F))F+\"\"$F)!\"\"*(\"$W\"F))F+\"\"%F)%#y3GF)F4*&\"#;F))F0F8F)F4 **\"#%)F)F+F)F/F)F9F)F)*(\"$N\"F))F+F1F))F9F1F)F)*(\"$3\"F)F2F)%\"yGF) F)*&F(F))F9F3F)F4*(F(F)F/F)FEF)F)**\"#aF)F9F)F+F)FEF)F4*&\"#\")F))FEF1 F)F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "OthSep := SepPoly(e ta2,[I2,I3,I4],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'OthSepG<#,8*& \"#O\"\"\")%#y1G\"\"'F)F)*(\"$g\"F))%#y2G\"\"#F))F+\"\"$F)!\"\"*(\"$W \"F))F+\"\"%F)%#y3GF)F4*&\"#;F))F0F8F)F4**\"#%)F)F+F)F/F)F9F)F)*(\"$N \"F))F+F1F))F9F1F)F)*(\"$3\"F)F2F)%\"yGF)F)*&F(F))F9F3F)F4*(F(F)F/F)FE F)F)**\"#aF)F9F)F+F)FEF)F4*&\"#\")F))FEF1F)F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "# Notice these two polynomials are the same and in fact that is because eta and eta2 are it's two roots with respect \+ to y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eval(Sep,[y=HIGHLIGHT]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<#,8*&\"#O\"\"\")%#y1G\"\"'F'F'*(\"$g\"F')%#y2G\"\"#F ')F)\"\"$F'!\"\"*(\"$W\"F')F)\"\"%F'%#y3GF'F2*&\"#;F')F.F6F'F2**\"#%)F 'F)F'F-F'F7F'F'*(\"$N\"F')F)F/F')F7F/F'F'*(\"$3\"F'F0F'%*HIGHLIGHTGF'F '*&F&F')F7F1F'F2*(F&F'F-F'FCF'F'**\"#aF'F7F'F)F'FCF'F2*&\"#\")F')FCF/F 'F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "GB:=gbasis([eta,I2,I 3,I4],[x1,x2,x3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GBG7&*(%#x1G \"\"\")%#x2G\"\"#F()%#x3G\"\"$F(*$)F-\"\"%F(,**$)F*F.F(F(*&F)F(F-F(!\" \"*&F*F()F-F+F(F(*$F,F(F6,.*$)F'F+F(F(*$F)F(F(*$F8F(F(*&F'F(F*F(F6*&F' F(F-F(F(*&F*F(F-F(F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "nor malf(eta2,GB,[x1,x2,x3]); # So eta2 is in the ideal generated by eta, I2, I3, and I4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "# Moral: one ETA is indeed sufficient and furthermor e, eta^2 is a linear combination of powers of I2, I3, I4, and eta^1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Part (8) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 444 "# Any invariant of R^(NG) is generated by I2, I3, I4, eta, and et a2.\n# From part (7) we know that\n# i) I2, I3, I4, and eta are eno ugh to generate\n# ii) eta^2 = linear combination of eta^1, and powe rs of I2, I3, I4\n# Thus we can rewrite any higher power of eta as low er order terms and any element of ring of invariants\n# can be written as linear combination of a polynomial in I2, I3, I4 and eta^1 multipl ied by a polynomial in I2, I3, I4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 16 "Further Examples" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 " Questions from Winter Final 2003" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 "1.5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "F := pro c(la, n) sort(simplify(evalsf(s[op(la)], (q^n-1)/(q-1) )),q); end;\nG \+ := proc(la) sort(simplify(evalsf(s[op(la)], 1/(1-q) )),q); end;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6$%#laG%\"nG6\"F)F)-%%sortG6$- %)simplifyG6#-%'evalsfG6$&%\"sG6#-%#opG6#9$*&,&)%\"qG9%\"\"\"F?!\"\"F? ,&F=F?F?F@F@F=F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGf*6#%#la G6\"F(F(-%%sortG6$-%)simplifyG6#-%'evalsfG6$&%\"sG6#-%#opG6#9$*&\"\"\" F:,&F:F:%\"qG!\"\"F=F " 0 "" {MPLTEXT 1 0 83 "M := proc(A, k) local i, r; r := product((1-q^i), i=1..k); sort( expand(A*r)); end; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGf*6$%\"AG %\"kG6$%\"iG%\"rG6\"F,C$>8%-%(productG6$,&\"\"\"F4)%\"qG8$!\"\"/F7;F49 %-%%sortG6#-%'expandG6#*&9$F4F/F4F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sort(F( [1,2],2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"qG\"\"#\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := F([2,1],2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*$)%\"qG\"\"#\"\"\"F*F(F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"qG\"\"\",&F$F%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "g := sort(expand(simplify((F([1,2],15))/ (q^2*( 1+q^2+q^4+q^6+q^8+q^10+q^12)*(1+q^2+q^4+q^6+q^8+q^10+q^12+q^14)*(q^4+q ^3+q^2+q+1)^2 ) )));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,D*$)%\"qG\"#=\"\"\"F**$)F(\"# " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "factor(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&,(*$)%\"qG\"\"#\"\"\"F)F'F)F)F)F)),0*$)F'\"\")F)F)*$)F'\"\"(F)!\"\"* $)F'\"\"&F)F)*$)F'\"\"%F)F2*$)F'\"\"$F)F)F'F2F)F)F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "sort(expand( (q^2+q+1)*(q^8-q^7+q^5 -q^4+q^3-q+1)^2*(q^4+q^3+q^2+q+1) ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,@*$)%\"qG\"#A\"\"\"F(*$)F&\"#>F(F(*$)F&\"# " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 "1.6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "H := \+ proc(n) local m, i, L, S; L := Par(n); m := nops(L); S := 0; for i fro m 1 to m do; S := S + s[op(L[i])]; od; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "seq(top(H(n)),n=1..10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,%#p1G*$)F#\"\"#\"\"\",&*(F&F'\"\"$!\"\"F#F*F'*&F*F+%#p3 GF'F',(*&#\"\"&\"#7F'*$)F#\"\"%F'F'F'*&#F'F*F'*&F-F'F#F'F'F'*&#F'F5F'* $)%#p2GF&F'F'F',**(\"#8F'\"#gF+F#F1F'*(F*F+F-F'F#F&F'*(F5F+F#F'F=F&F'* &F1F+%#p5GF'F',,*(\"#>F'\"$!=F+F#\"\"'F'**F&F'\"\"*F+F-F'F#F*F'*(F5F+F =F&F#F&F'*(F1F+FEF'F#F'F'*(F&F'FLF+F-F&F',0*(\"#HF'\"$I'F+F#\"\"(F'**F 1F'\"#OF+F-F'F#F5F'*(FJF+F=F&F#F*F'*(F1F+FEF'F#F&F'**F&F'FLF+F-F&F#F'F '*(F2F+F-F'F=F&F'*&FTF+%#p7GF'F',6*(\"$\">F'\"&!35F+F#\"\")F'**F@F'FIF +F-F'F#F1F'**F1F'\"#[F+F=F&F#F5F'**F&F'\"#:F+FEF'F#F*F'**F&F'FLF+F-F&F #F&F'**F2F+F-F'F#F'F=F&F'*(FTF+FfnF'F#F'F'*&\"#KF+F=F5F'*(F`oF+FEF'F-F 'F'*&F[oF+%#p4GF&F',<*(\"$J\"F'\"&W\"=F+F#FLF'**FHF'\"$S&F+F-F'F#FJF'* *F@F'\"$S#F+F=F&F#F1F'*(F2F+FEF'F#F5F'**F5F'\"#FF+F-F&F#F*F'**F2F+F-F' F=F&F#F&F'*(FTF+FfnF'F#F&F'*(FeoF+F#F'F=F5F'**F`oF+FEF'F-F'F#F'F'*(F[o F+F#F'FhoF&F'*(\"#?F+FEF'F=F&F'*(F1F'\"#\")F+F-F*F'*&FLF+%#p9GF'F',B*( \"%(=\"F'\"'+OXF+F#\"#5F'**FRF'\"%!*=F+F-F'F#FTF'**FHF'\"$?(F+F=F&F#FJ F'**F@F'\"$+$F+FEF'F#F1F'**F1F'\"#aF+F-F&F#F5F'**\"#=F+F-F'F=F&F#F*F'* *F&F'\"#@F+FfnF'F#F*F'*(FeoF+F=F5F#F&F'**F`oF+FEF'F-F'F#F&F'*(F[oF+Fho F&F#F&F'**FjpF+FEF'F#F'F=F&F'**F1F'F\\qF+F-F*F#F'F'*(FLF+F#F'F^qF'F'*( F]rF+F-F&F=F&F'*(F_rF+FfnF'F-F'F'*(F*F'\"#DF+FEF&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 " 1.7" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "squaresig:=proc(sig) \nlocal out,i,n;\nn:=nops(sig);\nout:=[seq(0,i=1..n)];\nfor i from 1 t o n do\n out[i]:=sig[sig[i]];\n od;\nout;\nend:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 127 "Theta := proc(n) local L, S, i; L := permut e(n); S := 0; for i from 1 to n! do; S := S + x[op(squaresig(L[i]))]; \+ od; end; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ThetaGf*6#%\"nG6% %\"LG%\"SG%\"iG6\"F,C%>8$-_%)combinatG%(permuteG6#9$>8%\"\"!?(8&\"\"\" F;-%*factorialGF4%%trueG>F7,&F7F;&%\"xG6#-%#opG6#-%*squaresigG6#&F/6#F :F;F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Theta(3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"%\"\"\"&%\"xG6%F&\"\"#\"\"$F&F &&F(6%F+F&F*F&&F(6%F*F+F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Theta(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,:*&\"#5\"\"\"&%\"xG6& F&\"\"#\"\"$\"\"%F&F&&F(6&F&F,F*F+F&&F(6&F&F+F,F*F&&F(6&F+F&F*F,F&*&F* F&&F(6&F+F,F&F*F&F&*&F*F&&F(6&F,F+F*F&F&F&&F(6&F,F&F+F*F&&F(6&F*F+F&F, F&&F(6&F,F*F&F+F&*&F*F&&F(6&F*F&F,F+F&F&&F(6&F*F,F+F&F&&F(6&F+F*F,F&F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "fourierTheta:=proc(n) \nlocal pars,la,out;\npars:=Par(n);\nout:=NULL;\nfor la in pars do\n \+ out:=out,[la,fouriercoe(Theta(n),la)];\n od;\n[out];\nend:\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fourierTheta(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$7#\"\"$K%'matrixG6#7#7#\"\"'Q*pprint5516 \"7$7$\"\"#\"\"\"KF(6#7$7$F&\"\"!7$F7F&Q*pprint552F.7$7%F2F2F2KF(F)Q*p print553F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fourierTheta( 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'7$7#\"\"%K%'matrixG6#7#7#\"#C Q*pprint5546\"7$7$\"\"$\"\"\"KF(6#7%7%\"\")\"\"!F87%F8F7F87%F8F8F7Q*pp rint555F.7$7$\"\"#F>KF(6#7$7$\"#7F87$F8FCQ*pprint556F.7$7%F>F2F2KF(F4Q *pprint557F.7$7&F2F2F2F2KF(F)Q*pprint558F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "# fourierTheta(5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "invele mentTheta:=proc(n)\nlocal FT,i,out,IFT,la,coe;\nFT:=fourier(Theta(n),n );\nout:=NULL;\nfor i from 1 to nops(FT) do\n la:=FT[i][1];\n coe: =FT[i][2];\n out:=out,[la,inverse(coe)];\n od;\nIFT:=[out];\nend: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "invfourier:=proc(coe s)\nlocal out,la,AA,BB,i,HLA;\nout:=0;\nfor i from 1 to nops(coes) do \n la:=coes[i][1];\n BB:=coes[i][2];\n AA:=gaAA(la);\n HLA:=hla(la );\n out:=out+dotprod(AA,BB)/HLA;\n od;\nout;\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "invfourier(fourierTheta(3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"%\"\"\"&%\"xG6%F&\"\"#\"\"$F&F &&F(6%F+F&F*F&&F(6%F*F+F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fourierTheta(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$7#\"\"$K%' matrixG6#7#7#\"\"'Q*pprint5596\"7$7$\"\"#\"\"\"KF(6#7$7$F&\"\"!7$F7F&Q *pprint560F.7$7%F2F2F2KF(F)Q*pprint561F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "invelementTheta(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7%7$7#\"\"$K%'matrixG6#7#7##\"\"\"\"\"'Q*pprint5626\"7$7$\"\"#F-KF(6 #7$7$#F-F&\"\"!7$F9F8Q*pprint563F07$7%F-F-F-KF(F)Q*pprint564F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "invfourier(invelementTheta(3 ));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"&\"#=\"\"\"&%\"xG6%F (\"\"#\"\"$F(F(*&#F(F'F(&F*6%F,F-F(F(!\"\"*&#F(F'F(&F*6%F-F(F,F(F2" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "invfourier(fourierTheta(4)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,:*&\"#5\"\"\"&%\"xG6&F&\"\"#\"\" $\"\"%F&F&&F(6&F&F,F*F+F&&F(6&F&F+F,F*F&&F(6&F+F&F*F,F&*&F*F&&F(6&F+F, F&F*F&F&*&F*F&&F(6&F,F+F*F&F&F&&F(6&F,F&F+F*F&&F(6&F*F+F&F,F&&F(6&F,F* F&F+F&*&F*F&&F(6&F*F&F,F+F&F&&F(6&F*F,F+F&F&&F(6&F+F*F,F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fourierTheta(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'7$7#\"\"%K%'matrixG6#7#7#\"#CQ*pprint5656 \"7$7$\"\"$\"\"\"KF(6#7%7%\"\")\"\"!F87%F8F7F87%F8F8F7Q*pprint566F.7$7 $\"\"#F>KF(6#7$7$\"#7F87$F8FCQ*pprint567F.7$7%F>F2F2KF(F4Q*pprint568F. 7$7&F2F2F2F2KF(F)Q*pprint569F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "invelementTheta(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'7$7# \"\"%K%'matrixG6#7#7##\"\"\"\"#CQ*pprint5706\"7$7$\"\"$F-KF(6#7%7%#F- \"\")\"\"!F:7%F:F8F:7%F:F:F8Q*pprint571F07$7$\"\"#F@KF(6#7$7$#F-\"#7F: 7$F:FEQ*pprint572F07$7%F@F-F-KF(F5Q*pprint573F07$7&F-F-F-F-KF(F)Q*ppri nt574F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "invfourier(invel ementTheta(4));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,:*&#\"\"\"\"$)GF &&%\"xG6&\"\"#\"\"$F&\"\"%F&!\"\"*&#F&F'F&&F)6&F+F-F,F&F&F.*&#F&F'F&&F )6&F,F+F-F&F&F.*&#F&\"#sF&&F)6&F,F-F&F+F&F.*&#F&F9F&&F)6&F+F&F-F,F&F.* &#F&F'F&&F)6&F,F&F+F-F&F.*&#F&\"\"*F&&F)6&F&F+F,F-F&F&*&#F&F'F&&F)6&F& F,F-F+F&F.*&#F&F'F&&F)6&F-F+F&F,F&F.*&#F&F9F&&F)6&F-F,F+F&F&F.*&#F&F'F &&F)6&F-F&F,F+F&F.*&#F&F'F&&F)6&F&F-F+F,F&F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "# invfourier(fourierTheta(5));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "# fourierTheta(5);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "# invelementTheta(5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "# invfourier(invelementTheta(5));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 "1.8" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "# Examples of \+ the four kinds of nontrival rotations of faces of octahedron (can also think of permuting the vertices of the cube since octahedron is its d ual)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "alpha13 := (1342,5786); #6 of them" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(alpha13G6$\"%U8\"%'y&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "beta12 := (1,253,467,8); #8 of them " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'beta12G6&\"\"\"\"$`#\"$n%\"\") " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Gamma := (17,24,38,56); #6 of them" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GammaG6&\"#<\"#C\"#Q \"#c" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "alpha2 := (14,23,58 ,67); #3 of them" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'alpha2G6&\"#9\" #B\"#e\"#n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Ch := (p1^8+6*p4^2+8*p1^2*p3^2+9*p2 ^4);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Chii := Ch/24;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ChG,**$)%#p1G\"\")\"\"\"F**&\"\"'F*)%#p4G\" \"#F*F**(F)F*)F(F/F*)%#p3GF/F*F**&\"\"*F*)%#p2G\"\"%F*F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%ChiiG,**&\"#C!\"\"%#p1G\"\")\"\"\"*&\"\"%F(%# p4G\"\"#F+*(\"\"$F(F)F/%#p3GF/F+*(F1F+F*F(%#p2GF-F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tom(Chii);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,N*&\"\"$\"\"\"&%\"mG6$\"\"&F%F&F&*&\"\"(F&&F(6%F*\"\"#F&F&F&*& \"#9F&&F(6&F*F&F&F&F&F&*&F,F&&F(6$\"\"%F7F&F&*&\"#8F&&F(6%F7F%F&F&F&*& \"#AF&&F(6%F7F/F/F&F&*&\"#NF&&F(6&F7F/F&F&F&F&*&\"#qF&&F(6'F7F&F&F&F&F &F&*&\"#CF&&F(6%F%F%F/F&F&*&\"#[F&&F(6&F%F%F&F&F&F&*&\"$S\"F&&F(6'F%F/ F&F&F&F&F&*&FEF&&F(6&F%F/F/F&F&F&*&\"$!GF&&F(6(F%F&F&F&F&F&F&F&*&\"$9 \"F&&F(6&F/F/F/F/F&F&*&\"$?%F&&F(6(F/F/F&F&F&F&F&F&*&\"$5#F&&F(6'F/F/F /F&F&F&F&*&\"$S)F&&F(6)F/F&F&F&F&F&F&F&F&*&\"%!o\"F&&F(6*F&F&F&F&F&F&F &F&F&F&&F(6#\"\")F&&F(6$F,F&F&*&F%F&&F(6$\"\"'F/F&F&*&F%F&&F(6%FapF&F& F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "# These numbers count the number of inequi valent colorings using the appropriate partition assuming a sufficient number of colors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "RB := expand(evalsf(Chii,R+B ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RBG,4*$)%\"RG\"\")\"\"\"F**$ )%\"BGF)F*F**&F(F*)F-\"\"(F*F**&)F(F0F*F-F*F**(\"\"$F*)F(\"\"'F*)F-\" \"#F*F**(F4F*)F(\"\"&F*)F-F4F*F**(F0F*)F(\"\"%F*)F-F?F*F**(F4F*)F(F4F* )F-F;F*F**(F4F*)F(F8F*)F-F6F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "RBY := sort(expand(evalsf(Chii,R+B+Y)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RBYG,fp*$)%\"BG\"\")\"\"\"F**&)F(\"\"(F*%\"RGF*F**&F ,F*%\"YGF*F**(\"\"$F*)F(\"\"'F*)F.\"\"#F*F***F2F*F3F*F.F*F0F*F**(F2F*F 3F*)F0F6F*F**(F2F*)F(\"\"&F*)F.F2F*F***F-F*F;F*F5F*F0F*F***F-F*F;F*F.F *F9F*F**(F2F*F;F*)F0F2F*F**(F-F*)F(\"\"%F*)F.FDF*F***\"#8F*FCF*F=F*F0F *F***\"#AF*FCF*F5F*F9F*F***FGF*FCF*F.F*FAF*F**(F-F*FCF*)F0FDF*F**(F2F* )F(F2F*)F.F " 0 "" {MPLTEXT 1 0 43 "RBYG := sort(expand(evalsf(Chii,R+B+Y+G))); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%RBYGG,f_l*$)%\"BG\"\")\"\"\"F** &)F(\"\"(F*%\"RGF*F**&F,F*%\"YGF*F**&F,F*%\"GGF*F**(\"\"$F*)F(\"\"'F*) F.\"\"#F*F***F4F*F5F*F.F*F0F*F***F4F*F5F*F.F*F2F*F**(F4F*F5F*)F0F8F*F* **F4F*F5F*F0F*F2F*F**(F4F*F5F*)F2F8F*F**(F4F*)F(\"\"&F*)F.F4F*F***F-F* FAF*F7F*F0F*F***F-F*FAF*F7F*F2F*F***F-F*FAF*F.F*F " 0 "" {MPLTEXT 1 0 17 "70*G^2*R^3*B^2*Y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"#q\"\" \")%\"BG\"\"#F&)%\"RG\"\"$F&%\"YGF&)%\"GGF)F&F&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "# Notice that RB, RBY, RBYG are the monomial expansions truncated acc ording to the number of distinct colors. " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tos(C hii);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,@&%\"sG6#\"\")\"\"\"*&\"\"#F (&F%6$\"\"'F*F(F(*&F*F(&F%6%\"\"&F*F(F(F(*&\"\"$F(&F%6&F1F(F(F(F(F(*& \"\"%F(&F%6$F7F7F(F(*&F*F(&F%6%F7F3F(F(F(*&F1F(&F%6%F7F*F*F(F(*&F*F(&F %6&F7F*F(F(F(F(*&F3F(&F%6'F7F(F(F(F(F(F(*&F1F(&F%6&F3F3F(F(F(F(*&F*F(& F%6&F3F*F*F(F(F(*&F*F(&F%6'F3F*F(F(F(F(F(*&F7F(&F%6&F*F*F*F*F(F(*&F*F( &F%6(F*F*F(F(F(F(F(F(&F%6*F(F(F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "to m(Chii);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,N*&\"\"$\"\"\"&%\"mG6$\" \"'\"\"#F&F&&F(6$\"\"(F&F&&F(6#\"\")F&*&F%F&&F(6%F*F&F&F&F&*&F.F&&F(6% \"\"&F+F&F&F&*&F%F&&F(6$F8F%F&F&*&\"#9F&&F(6&F8F&F&F&F&F&*&\"#8F&&F(6% \"\"%F%F&F&F&*&F.F&&F(6$FDFDF&F&*&\"#AF&&F(6%FDF+F+F&F&*&\"#NF&&F(6&FD F+F&F&F&F&*&\"#qF&&F(6'FDF&F&F&F&F&F&*&\"#CF&&F(6%F%F%F+F&F&*&\"#[F&&F (6&F%F%F&F&F&F&*&\"$S\"F&&F(6'F%F+F&F&F&F&F&*&FQF&&F(6&F%F+F+F&F&F&*& \"$!GF&&F(6(F%F&F&F&F&F&F&F&*&\"$9\"F&&F(6&F+F+F+F+F&F&*&\"$5#F&&F(6'F +F+F+F&F&F&F&*&\"$?%F&&F(6(F+F+F&F&F&F&F&F&*&\"$S)F&&F(6)F+F&F&F&F&F&F &F&F&*&\"%!o\"F&&F(6*F&F&F&F&F&F&F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Cube \+ := 1/24*(p1^6 + 6*p1^2*p4+3*p1^2*p2^2+8*p3^2+6*p2^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%CubeG,,*&\"#C!\"\"%#p1G\"\"'\"\"\"*(\"\"%F(%#p4 GF+F)\"\"#F+*(\"\")F(%#p2GF/F)F/F+*&\"\"$F(%#p3GF/F+*&F-F(F2F4F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tos(Cube);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*&%\"sG6#\"\"'\"\"\"&F%6$\"\"%\"\"#F(&F%6&\"\"$F(F( F(F(*&F,F(&F%6%F,F,F,F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tom(Cube);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8&%\"mG6#\"\"'\"\" \"*&\"\"#F(&F%6$\"\"$F-F(F(*&\"\"&F(&F%6&F-F(F(F(F(F(*&F-F(&F%6%F-F*F( F(F(*&F*F(&F%6%\"\"%F(F(F(F(*&F'F(&F%6%F*F*F*F(F(&F%6$F/F(F(*&\"\")F(& F%6&F*F*F(F(F(F(*&F*F(&F%6$F8F*F(F(*&\"#:F(&F%6'F*F(F(F(F(F(F(*&\"#IF( &F%6(F(F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 125 " Part(1) and (2) of Qual 2002 problem R1 redone ass uming D6 acted on Q[x1,x2,x3] by D6 acting on \{x1,x2\} and leaving x3 fixed" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 736 "DD6 := [matrix([[ 1, 0, 0], [0, 1, 0], [0, 0, 1]]), matrix([[1/2, -1/2*sqrt(3), 0], [1/2 *sqrt(3), 1/2, 0], [0, 0, 1]]), matrix([[-1/2, -1/2*sqrt(3), 0], [1/2* sqrt(3), -1/2, 0], [0, 0, 1]]), matrix([[-1, 0, 0], [0, -1, 0], [0, 0, 1]]), matrix([[-1/2, 1/2*sqrt(3), 0], [-1/2*sqrt(3), -1/2, 0], [0, 0, 1]]), matrix([[1/2, 1/2*sqrt(3), 0], [-1/2*sqrt(3), 1/2, 0], [0, 0, 1 ]]), matrix([[1, 0, 0], [0, -1, 0], [0, 0, 1]]), matrix([[1/2, -1/2*sq rt(3), 0], [-1/2*sqrt(3), -1/2, 0], [0, 0, 1]]), matrix([[-1/2, -1/2*s qrt(3), 0], [-1/2*sqrt(3), 1/2, 0], [0, 0, 1]]), matrix([[-1, 0, 0], [ 0, 1, 0], [0, 0, 1]]), matrix([[-1/2, 1/2*sqrt(3), 0], [1/2*sqrt(3), 1 /2, 0], [0, 0, 1]]), matrix([[1/2, 1/2*sqrt(3), 0], [1/2*sqrt(3), -1/2 , 0], [0, 0, 1]])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DD6G7.K%'mat rixG6#7%7%\"\"\"\"\"!F,7%F,F+F,7%F,F,F+Q*pprint5756\"KF'6#7%7%#F+\"\"# ,$*&F6!\"\"\"\"$F5F9F,7%,$*&F6F9F:F5F+F5F,F.Q*pprint576F0KF'6#7%7%#F9F 6F7F,7%F " 0 "" {MPLTEXT 1 0 23 "hilte st := hilb(DD6,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(hiltestG,,*& \"\"\"F'*&\"#7F',*F'F'*&\"\"$F'%\"qGF'!\"\"*&F,F')F-\"\"#F'F'*$)F-F,F' F.F'F.F'*&F'F'*&\"\"'F',*F'F'*&F1F'F-F'F.*&F1F'F0F'F'F2F.F'F.F'*&F'F'* &F6F',&F'F'F2F.F'F.F'*&F'F'*&F)F',*F'F'F-F'*$F0F'F.F2F.F'F.F'*&F'F'*&F 1F',*F2F'F@F.F-F.F'F'F'F.F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "factor(simplify(hiltest));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*& \"\"\"F%**),&%\"qGF%F%!\"\"\"\"$F%),&F)F%F%F%\"\"#F%,(*$)F)F.F%F%F)F%F %F%F%,(F0F%F)F*F%F%F%F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "realhil := 1/(1-q^6)/(1-q^2)/(1-q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(realhilG*&\"\"\"F&*(,&F&F&*$)%\"qG\"\"'F&!\"\"F&,&F&F&*$)F+\" \"#F&F-F&,&F&F&F+F-F&F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " factor(simplify(realhil-hiltest));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "# So here Hilbert \+ polynomial has three factors" }}}{EXCHG {PARA 3 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 " Invertibility example s" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "invertibility of (1,2)+(2,3 )+(3,4)+(4,5)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "make1:=pro c(la)\nglobal M12,M23,M34,M45,temp,out;\nprep(la);\nM12:=NAT([2,1,3,4, 5]);\nM23:=NAT([1,3,2,4,5]);\nM34:=NAT([1,2,4,3,5]);\nM45:=NAT([1,2,3, 5,4]);\ntemp:=matadd(M12,M23);\ntemp:=matadd(temp,M34);\nout:=matadd(t emp,M45);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "M:=make1 ([3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGK%'matrixG6#7'7'!\"# \"\"!!\"\"F+\"\"$7'\"\"\"F+F+F+F*7'F/F+\"\"#F/F,7'F+F/F/F/F/7'F+F+F+F/ F-Q*pprint5876\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(M); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "invertibi lity of (1,2)+(2,3)+(3,4)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "make2:=proc(la)\nglobal M12,M23,M34,temp,out;\nprep(la);\nM12:=NA T([2,1,3,4]);\nM23:=NAT([1,3,2,4]);\nM34:=NAT([1,2,4,3]);\ntemp:=matad d(M12,M23);\nout:=matadd(temp,M34);\nend:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "M:=make2([3,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"MGK%'matrixG6#7%7%\"\"!F*!\"\"7%\"\"\"F-F-7%F*F-\"\"#Q*pprint5886 \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(M);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "invertibility of (1,2)+ (2,3)+(3,4)+(4,5)+(5,6)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 310 " make3:=proc(la)\nglobal M12,M23,M34,M45,M56,temp,out;\nprep(la);\nM12: =NAT([2,1,3,4,5,6]);\nM23:=NAT([1,3,2,4,5,6]);\nM34:=NAT([1,2,4,3,5,6] );\nM45:=NAT([1,2,3,5,4,6]);\nM56:=NAT([1,2,3,4,6,5]);\ntemp:=matadd(M 12,M23);\ntemp:=matadd(temp,M34);\ntemp:=matadd(temp,M45);\ntemp:=mata dd(temp,M56);\nout:=matadd(temp,M45);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "M:=make3([3,1,1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGK%'matrixG6#7,7,!\"#\"\"#\"\"!F,F,F,!\"\"F-F,F,7,F+F*\"\" \"F/F,F,F/F,F-F,7,F,F/F,F,F/F,F,F/F/F,7,F,F/F,!\"$F/F,F,F,F,F-7,F,F,F/ F/F-F+F,F,F,F/7,F,F,F,F,F+F-F,F,F,F-7,F,F,F,F/F,F,F*F/F,F,7,F,F,F,F,F/ F,F/F,F+F,7,F,F,F,F,F,F/F,F+F,F/7,F,F,F,F,F,F,F,F,F/F-Q*pprint5896\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(M);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#!#!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 " the powers of x2,x3,x4 for n=4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "X2:=h2-(x[3]+x[4])*e1+x[3]*x[4]-(e1-x[2]-x[3] -x[4])*(e1-x[3]-x[4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X2G,*%#h2 G\"\"\"*&,&&%\"xG6#\"\"$F'&F+6#\"\"%F'F'%#e1GF'!\"\"*&F.F'F*F'F'*&,*F1 F'&F+6#\"\"#F2F*F2F.F2F',(F1F'F*F2F.F2F'F2" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "boo := evalsf(X2,size([seq(x[i],i=1..4)]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$booG,4*&#\"\"\"\"\"#F(*$),*&%\"xG6# F(F(&F.6#F)F(&F.6#\"\"$F(&F.6#\"\"%F(F)F(F(!\"\"*&,(F0F(F2F(F5F(F(F,F( F(*&F5F(F2F(F8*&F0F(F2F(F8*&F5F(F0F(F8*&#F(F)F(*$)F2F)F(F(F8*&#F(F)F(* $)F5F)F(F(F8*&#F(F)F(*$)F-F)F(F(F(*&FGF(*$)F0F)F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "foo := factor(boo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fooG*$)&%\"xG6#\"\"#F*\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "expand(evalsf(foo,x[1]+x[2]+x[3]+x[4])); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)&%\"xG6#\"\"#F(\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "X3:=-x[3]^2*x[4]-x[3]*x[4]^ 2-x[4]^3+h3-(e1-x[3]-x[4])*h2\n+x[2]*e1^2-X2*e1-x[2]*x[3]*e1-x[2]*x[4] *e1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X3G,4*&&%\"xG6#\"\"%\"\"\") &F(6#\"\"$\"\"#F+!\"\"*&)F'F0F+F-F+F1*$)F'F/F+F1%#h3GF+*&,(%#e1GF+F-F1 F'F1F+%#h2GF+F1*&&F(6#F0F+)F9F0F+F+*&,*F:F+*&,&F-F+F'F+F+F9F+F1*&F'F+F -F+F+*&,*F9F+F " 0 "" {MPLTEXT 1 0 8 "toe(X3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&,(*&&%\"xG6#\"\"%\"\"\"&F(6#\"\"$F+F+*$)F, \"\"#F+F+*$)F'F1F+F+F+%#e1GF+F+*&F'F+F0F+!\"\"*&F3F+F,F+F6*$)F'F.F+F6% #e3GF+*&,&F,F6F'F6F+%#e2GF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "booproc:=P->expand(evalsf(P,x[1]+x[2]+x[3]+x[4]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%(booprocGf*6#%\"PG6\"6$%)operatorG%&arrowGF(-% 'expandG6#-%'evalsfG6$9$,*&%\"xG6#\"\"\"F7&F56#\"\"#F7&F56#\"\"$F7&F56 #\"\"%F7F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "series4:= (1-x[1]*t)*(1-x[2]*t)*(1-x[3]*t)/(1-e1*t+e2*t^2-e3*t^3+e4*t^4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(series4G**,&\"\"\"F'*&&%\"xG6#F'F'% \"tGF'!\"\"F',&F'F'*&&F*6#\"\"#F'F,F'F-F',&F'F'*&&F*6#\"\"$F'F,F'F-F', ,F'F'*&%#e1GF'F,F'F-*&%#e2GF')F,F2F'F'*&%#e3GF')F,F7F'F-*&%#e4GF')F,\" \"%F'F'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "SoS := series( series4,t,5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$SoSG+/%\"tG\"\"\" \"\"!,*&%\"xG6#\"\"$!\"\"&F+6#\"\"#F.&F+6#F'F.%#e1GF'F',**&,&F/F.F2F.F 'F*F'F.*&F/F'F2F'F'%#e2GF.*&,*F*F'F/F'F2F'F4F.F'F4F'F.F1,**(F*F'F/F'F2 F'F.%#e3GF'*&F;F'F9F'F'*&,2*&F/F'F*F'F.*&F2F'F*F'F.F8F.F9F'*&F4F'F*F'F '*&F4F'F/F'F'*&F4F'F2F'F'*$)F4F1F'F.F'F4F'F.F-,*%#e4GF.*&F;F'F>F'F.*&F AF'F9F'F'*&,F.*&F9F'F*F'F.*&F9F'F/F'F.*&F9F'F2F'F.*(F1F'F4F'F9F 'F'*(F/F'F*F'F4F'F.*(F4F'F2F'F*F'F.*(F4F'F/F'F2F'F.*&FHF'F*F'F'*&F/F'F HF'F'*&FHF'F2F'F'*$)F4F-F'F.F'F4F'F.\"\"%-%\"OGF3\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "coeff(SoS,t,4);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,*%#e4G!\"\"*&,*&%\"xG6#\"\"$\"\"\"&F)6#\"\"#F,&F)6#F ,F,%#e1GF%F,%#e3GF,F%*&,2*&F-F,F(F,F%*&F0F,F(F,F%*&F-F,F0F,F%%#e2GF,*& F2F,F(F,F,*&F2F,F-F,F,*&F2F,F0F,F,*$)F2F/F,F%F,F9F,F,*&,<*(F(F,F-F,F0F ,F,F3F%*&F9F,F(F,F%*&F9F,F-F,F%*&F9F,F0F,F%*(F/F,F2F,F9F,F,*(F-F,F(F,F 2F,F%*(F2F,F0F,F(F,F%*(F2F,F-F,F0F,F%*&F>F,F(F,F,*&F-F,F>F,F,*&F>F,F0F ,F,*$)F2F+F,F%F,F2F,F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "X 4:=toe(coeff(SoS,t,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X4G,8*$) %#e1G\"\"%\"\"\"F**&,(&%\"xG6#\"\"$!\"\"&F.6#\"\"#F1&F.6#F*F1F*)F(F0F* F**&,(*&F2F*F-F*F**&F5F*F-F*F**&F2F*F5F*F*F*)F(F4F*F**(F0F*%#e2GF*F=F* F1**F(F*F-F*F2F*F5F*F1*(F4F*%#e3GF*F(F*F**(,(*&F4F*F-F*F**&F4F*F2F*F** &F4F*F5F*F*F*F?F*F(F*F**$)F?F4F*F**&,(F:F1F;F1F " 0 "" {MPLTEXT 1 0 39 "YY := subs(x[1]= e1-x[2]-x[3]-x[4],X4);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG,8*$ )%#e1G\"\"%\"\"\"F**&,&F(!\"\"&%\"xG6#F)F*F*)F(\"\"$F*F**&,(*&&F/6#\" \"#F*&F/6#F2F*F**&,*F(F*F6F-F9F-F.F-F*F9F*F**&F6F*FF*F-**F(F*F9F*F6F*F " 0 "" {MPLTEXT 1 0 8 "toe(YY);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,6*&,(&%\"xG6#\"\"#\"\"\"&F'6#\"\"$F*& F'6#\"\"%F*F*)%#e1GF-F*F**&,,*(F)F*F&F*F+F*!\"\"*$)F+F)F*F6*&F.F*F+F*F 6*$)F&F)F*F6*&F.F*F&F*F6F*)F2F)F*F**&%#e2GF*F=F*F6**F&F*F+F*F%F*F2F*F* *&%#e3GF*F2F*F**(,(F+F6F&F6*&F)F*F.F*F6F*F?F*F2F*F*%#e4GF6*&F.F*FBF*F* *&,,*&F&F*F+F*F*F7F*F9F*F:F*F " 0 "" {MPLTEXT 1 0 20 "X4:=expand(toe(YY));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X4G,R*&&%\"xG6#\"\"#\"\"\")%#e1G\"\"$F+F+*&F,F+&F(6# F.F+F+*&F,F+&F(6#\"\"%F+F+**F*F+F'F+F0F+)F-F*F+!\"\"*&F7F+)F0F*F+F8*(F 7F+F3F+F0F+F8*&F7F+)F'F*F+F8*(F7F+F3F+F'F+F8*&%#e2GF+F7F+F8*(F=F+F0F+F -F+F+*(F'F+F:F+F-F+F+**F'F+F0F+F-F+F3F+F+*&%#e3GF+F-F+F+*(F@F+F-F+F0F+ F8*(F@F+F-F+F'F+F8**F*F+F-F+F@F+F3F+F8%#e4GF8*&F3F+FEF+F+*(F@F+F'F+F0F +F+*&F@F+F:F+F+*(F@F+F3F+F0F+F+*&F@F+F=F+F+*(F@F+F3F+F'F+F+*$)F@F*F+F+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "cc2:=coeff(X4,x[2],2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cc2G,(*$)%#e1G\"\"#\"\"\"!\"\"* &F(F*&%\"xG6#\"\"$F*F*%#e2GF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "XX4:=X4-cc2*x[2]^2+cc2*X2;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% $XX4G,V**&%\"xG6#\"\"#\"\"\"&F(6#\"\"$F+%#e1GF+&F(6#\"\"%F+F+%#e4G!\" \"*&%#e3GF+F/F+F+*$)%#e2GF*F+F+*&F9F+)F/F*F+F4*&)F/F.F+F,F+F+*&F'F+F=F +F+*(F9F+F'F+F,F+F+*(F9F+F/F+F,F+F4*(F9F+F/F+F'F+F4**F*F+F'F+F,F+F;F+F 4*&F0F+F6F+F+*&F=F+F0F+F+*&F;F+)F,F*F+F4*(F;F+F0F+F,F+F4*(F;F+F0F+F'F+ F4*()F'F*F+F,F+F/F+F+*(F'F+FFF+F/F+F+**F*F+F/F+F9F+F0F+F4*(F9F+F0F+F,F +F+*(F9F+F0F+F'F+F+*&F9F+FFF+F+*&F9F+FJF+F+*&,(*$F;F+F4*&F/F+F,F+F+F9F +F+FJF+F4*&FRF+,*%#h2GF+*&,&F,F+F0F+F+F/F+F4*&F0F+F,F+F+*&,*F/F+F'F4F, F4F0F4F+,(F/F+F,F4F0F4F+F4F+F+*&F;F+FJF+F4" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "expand(XX4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,F% #e4G!\"\"*&%#e3G\"\"\"%#e1GF(F(*$)%#e2G\"\"#F(F(*(F-F(F,F()F)F-F(F%*&) F)\"\"$F(&%\"xG6#F2F(F%*$)F)\"\"%F(F(*&&F46#F8F(F'F(F(*&F/F()F3F-F(F(* (F/F(F:F(F3F(F(*(F)F(F,F(F:F(F%*(F)F(F=F(F:F(F%*(F)F(F3F()F:F-F(F%*(F) F(F3F(%#h2GF(F(*&F/F(FBF(F(*&F)F()F3F2F(F%*&F,F(FDF(F(*&F,F(FBF(F%*&F/ F(FDF(F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "X4+e1*x[3]^3-e1 *X3;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,V**&%\"xG6#\"\"#\"\"\"&F&6#\" \"$F)%#e1GF)&F&6#\"\"%F)F)%#e4G!\"\"*&%#e3GF)F-F)F)*$)%#e2GF(F)F)*&F7F ))F-F(F)F2*&)F-F,F)F*F)F)*&F%F)F;F)F)*(F7F)F%F)F*F)F)*(F7F)F-F)F*F)F2* (F7F)F-F)F%F)F2**F(F)F%F)F*F)F9F)F2*&F.F)F4F)F)*&F;F)F.F)F)*&F9F))F*F( F)F2*(F9F)F.F)F*F)F2*(F9F)F.F)F%F)F2*()F%F(F)F*F)F-F)F)*(F%F)FDF)F-F)F )**F(F)F-F)F7F)F.F)F2*(F7F)F.F)F*F)F)*(F7F)F.F)F%F)F)*&F7F)FDF)F)*&F7F )FHF)F)*&F-F))F*F,F)F)*&F-F),4*&F.F)FDF)F2*&)F.F(F)F*F)F2*$)F.F,F)F2%# h3GF)*&,(F-F)F*F2F.F2F)%#h2GF)F2*&F%F)F9F)F)*&,*FenF)*&,&F*F)F.F)F)F-F )F2*&F.F)F*F)F)*&,*F-F)F%F2F*F2F.F2F)FZF)F2F)F-F)F2*(F%F)F*F)F-F)F2*(F %F)F.F)F-F)F2F)F2*&F9F)FHF)F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Ere := expand(X4+e1*x[3]^3-e1*X3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$EreG,`o**&%\"xG6#\"\"#\"\"\"&F(6#\"\"$F+%#e1GF+&F(6#\"\"%F+F+ %#e4G!\"\"*&%#e3GF+F/F+F+*$)%#e2GF*F+F+*&F9F+)F/F*F+F4*(F*F+)F/F.F+F,F +F+*&F'F+F=F+F+*$)F/F2F+F4*(F9F+F'F+F,F+F+*(F9F+F/F+F,F+F4*(F9F+F/F+F' F+F4**F*F+F'F+F,F+F;F+F4*&F0F+F6F+F+*(F*F+F=F+F0F+F+*(F*F+F;F+)F,F*F+F 4**F*F+F;F+F0F+F,F+F4*(F;F+F0F+F'F+F4*()F'F*F+F,F+F/F+F+*(F'F+FHF+F/F+ F+**F*F+F/F+F9F+F0F+F4*(F9F+F0F+F,F+F+*(F9F+F0F+F'F+F+*(F/F+FHF+F0F+F+ *(F/F+F,F+)F0F*F+F+*(F/F+F,F+%#h2GF+F4*(F/F+FUF+F0F+F4*&F9F+FHF+F+*&F9 F+FLF+F+*&F;F+FSF+F4*&F/F+)F,F.F+F+*&F/F+)F0F.F+F+*&F/F+%#h3GF+F4*(F*F +F;F+FUF+F+*&F;F+FLF+F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "t oe(Ere);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*&,(&%\"xG6#\"\"#\"\"\"& F'6#\"\"$F*&F'6#\"\"%F*F*)%#e1GF-F*F**&,.*&F)F*)F+F)F*!\"\"*&F.F*F&F*F 7*$)F&F)F*F7*(F)F*F.F*F+F*F7*(F)F*F&F*F+F*F7*$)F.F)F*F7F*)F2F)F*F**&%# e2GF*F?F*F7*&,0*(F.F*F+F*F&F*F**$)F+F-F*F**&F&F*F6F*F**&F>F*F+F*F**$)F .F-F*F**&F:F*F+F*F**&F.F*F6F*F*F*F2F*F**(,&F&F7F.F7F*FAF*F2F*F*%#e4GF7 *&F.F*%#e3GF*F**&,,*&F&F*F+F*F**$F6F*F**&F.F*F+F*F*F9F*F8F*F*FAF*F**$) FAF)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "XXX4:=toe(Ere); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%XXX4G,4*&,(&%\"xG6#\"\"#\"\"\"& F)6#\"\"$F,&F)6#\"\"%F,F,)%#e1GF/F,F,*&,.*&F+F,)F-F+F,!\"\"*&F0F,F(F,F 9*$)F(F+F,F9*(F+F,F0F,F-F,F9*(F+F,F(F,F-F,F9*$)F0F+F,F9F,)F4F+F,F,*&%# e2GF,FAF,F9*&,0*(F0F,F-F,F(F,F,*$)F-F/F,F,*&F(F,F8F,F,*&F@F,F-F,F,*$)F 0F/F,F,*&F " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 " Misc. examples involving SF package" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "SP:=itensor(s[3,2,1],s[4,2]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SPG,&*&#\"\"\"\"\"&F(*&%#p5GF( %#p1GF(F(!\"\"*&#F(F)F(*$)F,\"\"'F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "tos(SP);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4&%\"sG6$ \"\"&\"\"\"F(*&\"\"#F(&F%6$\"\"%F*F(F(*&F*F(&F%6%F-F(F(F(F(&F%6$\"\"$F 3F(*&F3F(&F%6%F3F*F(F(F(*&F*F(&F%6&F3F(F(F(F(F(&F%6%F*F*F*F(*&F*F(&F%6 &F*F*F(F(F(F(&F%6'F*F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "SP2:=skew(s[3,1,1,0],s[4,4,2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SP2G,.*(\"#7!\"\"%#p2G\"\"#%#p1G\"\"$F(*&\"\"(F(%#p7 G\"\"\"F(*(F'F(%#p3GF0F+\"\"%F(*(\"#6F0\"$?%F(F+F.F0*(\"\"&F(%#p5GF0F+ F*F0*(F'F(F2F0F)F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tos( SP2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,&%\"sG6%\"\"%\"\"#\"\"\"F)& F%6&F'F)F)F)F)&F%6%\"\"$F.F)F)&F%6%F.F(F(F)&F%6&F.F(F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "toe(SP2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*(%#e3G\"\"\"%#e2GF&)%#e1G\"\"#F&F&**F*F&%#e4GF&F)F&F 'F&!\"\"*&)F%F*F&F)F&F-*&F)F&%#e6GF&F&*&%#e5GF&F'F&F&*&F,F&F%F&F&%#e7G F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "toh(SP2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*(%#h3G\"\"\"%#h2GF&)%#h1G\"\"#F&F&**F*F&%#h4 GF&F'F&F)F&!\"\"*&F)F&)F%F*F&F-*&%#h6GF&F)F&F&*&%#h5GF&F'F&F&*&F,F&F%F &F&%#h7GF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tom(SP2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,6*&\"#:\"\"\"&%\"mG6&\"\"#F*F*F&F&F&* &\"#?F&&F(6'\"\"$F&F&F&F&F&F&*&\"\"*F&&F(6&F/F*F&F&F&F&*&\"#KF&&F(6'F* F*F&F&F&F&F&*&\"#mF&&F(6(F*F&F&F&F&F&F&F&*&\"$K\"F&&F(6)F&F&F&F&F&F&F& F&F&*&F*F&&F(6%F/F/F&F&F&*&\"\"%F&&F(6%F/F*F*F&F&&F(6%FDF*F&F&*&F/F&&F (6&FDF&F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "SP3:=top ((e4+h4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SP3G,(*&#\"\"#\"\"$\" \"\"*&%#p1GF*%#p3GF*F*F**&#F*\"\"%F**$)%#p2GF(F*F*F**&#F*\"#7F**$)F,F0 F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tos(SP3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"sG6#\"\"%\"\"\"&F%6&F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tom(SP3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,&%\"mG6#\"\"%\"\"\"&F%6$\"\"#F+F(&F%6$\"\"$F(F(& F%6%F+F(F(F(*&F+F(&F%6&F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "SP4:=top((e5+h5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$SP4G,**(\"\"#\"\"\"\"\"&!\"\"%#p5GF(F(*(\"\"$F*%#p3GF(%#p1GF'F(*( \"\"%F*F/F(%#p2GF'F(*&\"#gF*F/F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tos(SP4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"sG6 #\"\"&\"\"\"&F%6'F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tom(SP4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0&%\"mG6#\"\"&\"\" \"&F%6%\"\"#F+F(F(&F%6$\"\"$F+F(&F%6$\"\"%F(F(&F%6%F.F(F(F(&F%6&F+F(F( F(F(*&F+F(&F%6'F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "toe(p5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&\"\"&\"\"\"%#e5GF &F&*(F%F&%#e1GF&%#e4GF&!\"\"*(F%F&%#e3GF&%#e2GF&F+*(F%F&F-F&)F)\"\"#F& F&*(F%F&)F.F1F&F)F&F&*(F%F&F.F&)F)\"\"$F&F+*$)F)F%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "m[3,2]+m[5]+m[4,1]+m[3,1,1]+m[2,2,1 ]+m[2,1,1,1]+2*m[1,1,1,1,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0&%\" mG6#\"\"&\"\"\"&F%6%\"\"#F+F(F(&F%6$\"\"$F+F(&F%6$\"\"%F(F(&F%6%F.F(F( F(&F%6&F+F(F(F(F(*&F+F(&F%6'F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "SP5:=(p1^6+8*p3^2+3*p2^2*p1^2)/12;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SP5G,(*&#\"\"\"\"#7F(*$)%#p1G\"\"'F(F(F(*&#\"\" #\"\"$F(*$)%#p3GF0F(F(F(*&#F(\"\"%F(*&)%#p2GF0F()F,F0F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tos(SP5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,2&%\"sG6#\"\"'\"\"\"&F%6$\"\"%\"\"#F(&F%6%F+F(F(F(*& F,F(&F%6$\"\"$F2F(F(&F%6&F2F(F(F(F(*&F,F(&F%6%F,F,F,F(F(&F%6&F,F,F(F(F (&F%6(F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tom( SP5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8&%\"mG6#\"\"'\"\"\"*&\"\"%F (&F%6$\"\"$F-F(F(*&\"#5F(&F%6&F-F(F(F(F(F(*&F'F(&F%6%F-\"\"#F(F(F(*&F- F(&F%6%F*F(F(F(F(*&\"\"*F(&F%6%F5F5F5F(F(&F%6$\"\"&F(F(*&\"#;F(&F%6&F5 F5F(F(F(F(*&F5F(&F%6$F*F5F(F(*&\"#IF(&F%6'F5F(F(F(F(F(F(*&\"#gF(&F%6(F (F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "SP6:=eva lsf(s[3,2,2],1/(1-q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SP6G,:*& \"\"\"F'*&\"$S#F'),&%\"qGF'F'!\"\"\"\"(F'F-F-*&F'F'*(F)F',&*$)F,\"\"#F 'F'F'F-F')F+\"\"&F'F-F-*&F'F'*(\"#CF',&*$)F,\"\"$F'F'F'F-F')F+\"\"%F'F -F'*&F'F'*(F9F',&*$)F,F?F'F'F'F-F')F+F=F'F-F'*&F'F'*(\"#[F')F1F4F'FEF' F-F-*&F'F'*(\"#5F',&*$)F,F6F'F'F'F-F')F+F4F'F-F-*&F'F'**\"#7F'F:F'F1F' FPF'F-F-*&F'F'*(\"#;F'F+F')F1F=F'F-F'*&F'F'**\"\")F'FBF'F+F'F1F'F-F'*& F'F'*(F9F'F:F'FIF'F-F-*&F'F'*(FLF'FMF'F1F'F-F-*&F'F'*(FSF'FBF'F:F'F-F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "series(SP6,q,16);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"qG\"\"\"\"\"'\"\"#\"\"(\"\"&\"\") \"\"*F+\"#<\"#5\"#G\"#6\"#Y\"#7\"#q\"#8\"$1\"\"#9\"$`\"\"#:-%\"OG6#F% \"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "SP7:=factor(SP6*qfa c(7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SP7G**,0*$)%\"qG\"\"'\"\" \"F+*$)F)\"\"&F+F+*$)F)\"\"%F+F+*$)F)\"\"$F+F+*$)F)\"\"#F+F+F)F+F+F+F+ ,(F5F+F)F+F+F+F+,(F5F+F)!\"\"F+F+F+F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "expand(SP7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8*$) %\"qG\"\"'\"\"\"F(*$)F&\"\"(F(F(*&\"\"#F()F&\"\"*F(F(*&F-F()F&\"\")F(F (*$)F&\"#;F(F(*$)F&\"#:F(F(*&F-F()F&\"#9F(F(*&F-F()F&\"#8F(F(*&\"\"$F( )F&\"#7F(F(*&F@F()F&\"#6F(F(*&F@F()F&\"#5F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 " Invar iant Theory examples" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "D5" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "D5:=mkdihedr(5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "FRD5:=factor(hilb(D5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%FRD5G,$*,\"\"%\"\"\",*\"\"#F(%\"qGF(*&F*F ()F+F*F(F(*&F+F(\"\"&#F(F*!\"\"F1,*F*F(F+F(*&F*F(F-F(F(F.F(F1,&F+F(F(F (F1,&F+F(F(F1!\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "map( expand,series(FRD5,q,20));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+I%\"qG \"\"\"\"\"!F%\"\"#F%\"\"%F%\"\"&F%\"\"'F%\"\"(F%\"\")F%\"\"*F'\"#5F%\" #6F'\"#7F%\"#8F'\"#9F'\"#:F'\"#;F'\"#-%\"OG6#F%\"#?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "II1:=REY(D5,x1^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$II1G,&*&\"\"#!\"\"%#x1GF'\"\"\"*&F'F(%#x2 GF'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "II2:=REY(D5,x1^5); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$II2G,(*&#\"\"&\"\")\"\"\"*&)%#x 1G\"\"$F*)%#x2G\"\"#F*F*!\"\"*&#F(\"#;F**&F-F*)F0\"\"%F*F*F**&#F*F5F** $)F-F(F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "GB:=gbasis( [II1-y1,II2-y2],[x1,x2,y1,y2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# GBG7%,,*&\"\"%\"\"\")%#x2G\"\"'F)F)*(\"#9F))F+F(F)%#y1GF)!\"\"*(\"#8F) )F+\"\"#F))F0F5F)F)*&F5F))F0\"\"$F)F1*(F(F)%#x1GF)%#y2GF)F),**(F(F)F;F )F/F)F)**F,F)F;F)F4F)F0F)F1*&F6F)F;F)F)*&F(F)F " 0 "" {MPLTEXT 1 0 14 "map(print ,GB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"\"%\"\"\")%#x2G\"\"'F&F &*(\"#9F&)F(F%F&%#y1GF&!\"\"*(\"#8F&)F(\"\"#F&)F-F2F&F&*&F2F&)F-\"\"$F &F.*(F%F&%#x1GF&%#y2GF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(\"\"% \"\"\"%#x1GF&)%#x2GF%F&F&**\"\"'F&F'F&)F)\"\"#F&%#y1GF&!\"\"*&)F.F-F&F 'F&F&*&F%F&%#y2GF&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%#x1G\"\" #\"\"\"F(*$)%#x2GF'F(F(*&F'F(%#y1GF(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "D6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "D6:=mkdihedr(6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "FRD6:=factor(hilb(D6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%FRD6G*&\"\"\"F&**),&%\"qGF&F&!\"\"\"\"#F&),&F*F&F&F&F,F&,(*$) F*F,F&F&F*F&F&F&F&,(F0F&F*F+F&F&F&F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "map(expand,series(FRD6,q,20));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"qG\"\"\"\"\"!F%\"\"#F%\"\"%F'\"\"'F'\"\")F'\"#5\" \"$\"#7F,\"#9F,\"#;F(\"#=-%\"OG6#F%\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "II1:=REY(D6,x1^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$II1G,&*&\"\"#!\"\"%#x1GF'\"\"\"*&F'F(%#x2GF'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "II2:=REY(D6,x1^6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$II2G,**&#\"\"*\"#K\"\"\"*$)%#x2G\"\"'F*F*F**&#\"#6F) F**$)%#x1GF.F*F*F**&#\"#:F)F**&)F4\"\"%F*)F-\"\"#F*F*F**&#\"#XF)F**&)F 4F " 0 "" {MPLTEXT 1 0 32 "GBD6:=g basis([II1,II2],[x1,x2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%GBD6G7 $*$)%#x2G\"\"'\"\"\",&*$)%#x1G\"\"#F*F**$)F(F/F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "GB2D6:=gbasis([II1-y1,II2-y2],[x1,x2,y1,y2] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GB2D6G7$,,*&\"\"%\"\"\")%#x2G \"\"'F)F)*(\"#7F))F+F(F)%#y1GF)!\"\"*(\"\"*F))F+\"\"#F))F0F5F)F)*&\"#6 F))F0\"\"$F)F1*&F(F)%#y2GF)F),(*$)%#x1GF5F)F)*$F4F)F)*&F5F)F0F)F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(print,GB2D6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"\"%\"\"\")%#x2G\"\"'F&F&*(\"#7F&)F(F%F &%#y1GF&!\"\"*(\"\"*F&)F(\"\"#F&)F-F2F&F&*&\"#6F&)F-\"\"$F&F.*&F%F&%#y 2GF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%#x1G\"\"#\"\"\"F(*$)%# x2GF'F(F(*&F'F(%#y1GF(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 59 "The subgroup <(1,2),(3,4)> of S4, under the rep'n A^(2, 1,1)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "GG4:=[[1,2,3,4],[2,1 ,3,4],[1,2,4,3],[2,1,4,3]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GG4G 7&7&\"\"\"\"\"#\"\"$\"\"%7&F(F'F)F*7&F'F(F*F)7&F(F'F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "prep([2,1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(Q*pprint59 06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "GG211:=map(NAT,GG4) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GG211G7&K%'matrixG6#7%7%\"\"\" \"\"!F,7%F,F+F,7%F,F,F+Q*pprint5916\"KF'6#7%7%!\"\"F,F+7%F,F5F5F.Q*ppr int592F0KF'6#7%F-F*7%F,F,F5Q*pprint593F0KF'6#7%F6F4F;Q*pprint594F0" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "map(jordan,GG211);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&K%'matrixG6#7%7%\"\"\"\"\"!F*7%F*F)F *7%F*F*F)Q*pprint5956\"KF%6#7%7%!\"\"F*F*F+7%F*F*F3Q*pprint596F.KF%F0Q *pprint597F.KF%F0Q*pprint598F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "FRG211:=factor(hilb(GG211));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'FRG211G,$*(,(*$)%\"qG\"\"#\"\"\"F,F*!\"\"F,F,F,,&F*F,F,F,!\"#,&F* F,F,F-!\"$F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "factor(FRG2 11-(1+q^3)/(1-q^2)^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series(FRG211,q,20);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+K%\"qG\"\"\"\"\"!\"\"$\"\"#F%F'\"\"' \"\"%F'\"\"&\"#5F)F)\"\"(\"#:\"\")F,\"\"*\"#@F,F.\"#6\"#G\"#7F1\"#8\"# O\"#9F3F.\"#X\"#;F6\"#<\"#b\"#=F8\"#>-%\"OG6#F%\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "II1:=REY(GG211,x1^2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$II1G,&*&\"\"#!\"\"%#x1GF'\"\"\"*&F'F(%#x2GF'F*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "II2:=REY(GG211,x1*x2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$II2G*&%#x1G\"\"\"%#x2GF'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "II3:=REY(GG211,x3^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$II3G,.*$)%#x3G\"\"#\"\"\"F**&#F*F)F **$)%#x1GF)F*F*F**&F/F*%#x2GF*!\"\"*&F/F*F(F*F**&F,F**$)F1F)F*F*F**&F1 F*F(F*F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eta:=REY(GG211, (x1+x2+x3)^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$etaG,.*&#\"\"$\" \"#\"\"\"*$)%#x1GF(F*F*!\"\"*&#F(F)F**$)%#x2GF(F*F*F.*&#F(F)F**&F-F*)F 3F)F*F*F**&F5F**&)F-F)F*F3F*F*F**(F(F*F:F*%#x3GF*F.*(F(F*F7F*F " 0 "" {MPLTEXT 1 0 12 "factor(eta);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*,\"\"$\"\"\"\"\"#!\"\",&%#x1GF(%#x2GF&F&,&F+F &F*F&F&,(F+F(*&F'F&%#x3GF&F&F*F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "GB:=gbasis([II1-y1,II2-y2,II3-y3,eta-y],[x1,x2,x3,y1, y2,y3,y]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#GBG70,(*$)%#x2G\"\"$ \"\"\"F+*(\"\"#F+F)F+%#y1GF+!\"\"*&%#x1GF+%#y2GF+F+,0*(\"\"'F+)F)F-F+% #x3GF+F+*(F*F+F.F+F1F+F/*(F*F+F)F+F.F+F/*(F5F+F7F+F.F+F/*(F*F+F1F+F2F+ F+*(F*F+F)F+F2F+F+%\"yGF/,4*(F*F+F.F+F1F+F/F=F/*(F*F+F1F+F2F+F+*(F*F+F )F+F2F+F/*(F*F+F)F+F.F+F+*(F5F+F)F+%#y3GF+F/*(F5F+F7F+F.F+F/*(F5F+F)F+ )F7F-F+F+*(F5F+F2F+F7F+F+,6*$)F7F*F+F+*&F7F+FDF+F/*&F.F+F1F+F/*&F1F+FD F+F+F0F+*&F)F+F.F+F+*&F)F+FDF+F/*&F)F+F2F+F/*&F7F+F.F+F/*&F2F+F7F+F+,4 *(F5F+F6F+F.F+F+*(F5F+F6F+F2F+F/*(\"#7F+F6F+FDF+F/*&F5F+)F.F-F+F/*(F5F +F2F+F.F+F+*(FXF+F.F+FDF+F+*&F1F+F=F+F+*&F)F+F=F+F/*(F-F+F7F+F=F+F+,:* *FXF+F.F+F)F+F7F+F+**FXF+F)F+F7F+F2F+F/*(FXF+FGF+F2F+F+*(FXF+F6F+FDF+F /*&F5F+FZF+F/*(FXF+F2F+F.F+F+*&F5F+)F2F-F+F/*(FXF+F.F+FDF+F+*(FXF+F2F+ FDF+F/FgnF+*(F*F+F)F+F=F+F/*(F-F+F7F+F=F+F+,.*(F5F+FGF+F.F+F+*(F5F+FGF +F2F+F+*(F5F+F.F+FDF+F/*(F5F+F2F+FDF+F/FgnF/FhnF/,8*(F5F+FZF+F1F+F+**F 5F+F.F+F1F+F2F+F/**F5F+F)F+F.F+F2F+F/*(F5F+F)F+FboF+F+**FXF+F.F+F1F+FD F+F/**FXF+F2F+F)F+FDF+F+*&F6F+F=F+F+**F-F+F)F+F7F+F=F+F/*(F-F+FGF+F=F+ F+*&F2F+F=F+F/*(F-F+FDF+F=F+F/,4*(F5F+F)F+FZF+F+**F5F+F.F+F1F+F2F+F/** F5F+F)F+F.F+F2F+F/*(F5F+F1F+FboF+F+**FXF+F)F+F.F+FDF+F/**FXF+F1F+F2F+F DF+F+FcpF/**F-F+F)F+F7F+F=F+F+FfpF+,:*(F5F+FboF+F7F+F/**F5F+F.F+F1F+FD F+F+**F5F+F)F+F.F+FDF+F/*(F-F+F6F+F=F+F/*(F5F+F7F+FZF+F+*&FGF+F=F+F/*& F.F+F=F+F+**F5F+F2F+F)F+FDF+F/**F-F+F)F+F7F+F=F+F+*&FDF+F=F+F+**F5F+F1 F+F2F+FDF+F+FfpF+,0*(\"#=F+F2F+FZF+F/*(F^rF+F.F+FboF+F/*&F^rF+)F2F*F+F +*$)F=F-F+F+*&F^rF+)F.F*F+F+*(\"#OF+FboF+FDF+F+*(FgrF+FZF+FDF+F/,(*$)F 1F-F+F+*$F6F+F+*&F-F+F.F+F/,&*&F1F+F)F+F+F2F/,.F.F+*$FGF+F+*&F1F+F7F+F +*&F)F+F7F+F/FDF/F2F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ma p(print,GB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%#x2G\"\"$\"\"\"F (*(\"\"#F(F&F(%#y1GF(!\"\"*&%#x1GF(%#y2GF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*(\"\"'\"\"\")%#x2G\"\"#F&%#x3GF&F&*(\"\"$F&%#y1GF&%# x1GF&!\"\"*(F,F&F(F&F-F&F/*(F%F&F*F&F-F&F/*(F,F&F.F&%#y2GF&F&*(F,F&F(F &F3F&F&%\"yGF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*(\"\"$\"\"\"%#y1G F&%#x1GF&!\"\"%\"yGF)*(F%F&F(F&%#y2GF&F&*(F%F&%#x2GF&F,F&F)*(F%F&F.F&F 'F&F&*(\"\"'F&F.F&%#y3GF&F)*(F1F&%#x3GF&F'F&F)*(F1F&F.F&)F4\"\"#F&F&*( F1F&F,F&F4F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6*$)%#x3G\"\"$\"\" \"F(*&F&F(%#y3GF(!\"\"*&%#y1GF(%#x1GF(F+*&F.F(F*F(F(*&F.F(%#y2GF(F(*&% #x2GF(F-F(F(*&F3F(F*F(F+*&F3F(F1F(F+*&F&F(F-F(F+*&F1F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*(\"\"'\"\"\")%#x2G\"\"#F&%#y1GF&F&*(F%F&F 'F&%#y2GF&!\"\"*(\"#7F&F'F&%#y3GF&F-*&F%F&)F*F)F&F-*(F%F&F,F&F*F&F&*(F /F&F*F&F0F&F&*&%#x1GF&%\"yGF&F&*&F(F&F7F&F-*(F)F&%#x3GF&F7F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,:**\"#7\"\"\"%#y1GF&%#x2GF&%#x3GF&F&* *F%F&F(F&F)F&%#y2GF&!\"\"*(F%F&)F)\"\"#F&F+F&F&*(F%F&)F(F/F&%#y3GF&F,* &\"\"'F&)F'F/F&F,*(F%F&F+F&F'F&F&*&F4F&)F+F/F&F,*(F%F&F'F&F2F&F&*(F%F& F+F&F2F&F,*&%#x1GF&%\"yGF&F&*(\"\"$F&F(F&F=F&F,*(F/F&F)F&F=F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(\"\"'\"\"\")%#x3G\"\"#F&%#y1GF&F&* (F%F&F'F&%#y2GF&F&*(F%F&F*F&%#y3GF&!\"\"*(F%F&F,F&F.F&F/*&%#x1GF&%\"yG F&F/*&%#x2GF&F3F&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8*(\"\"'\"\"\" )%#y1G\"\"#F&%#x1GF&F&**F%F&F(F&F*F&%#y2GF&!\"\"**F%F&%#x2GF&F(F&F,F&F -*(F%F&F/F&)F,F)F&F&**\"#7F&F(F&F*F&%#y3GF&F-**F3F&F,F&F/F&F4F&F&*&)F/ F)F&%\"yGF&F&**F)F&F/F&%#x3GF&F8F&F-*(F)F&)F:F)F&F8F&F&*&F,F&F8F&F-*(F )F&F4F&F8F&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*(\"\"'\"\"\"%#x2GF &)%#y1G\"\"#F&F&**F%F&F)F&%#x1GF&%#y2GF&!\"\"**F%F&F'F&F)F&F-F&F.*(F%F &F,F&)F-F*F&F&**\"#7F&F'F&F)F&%#y3GF&F.**F3F&F,F&F-F&F4F&F&*&)F'F*F&% \"yGF&F.**F*F&F'F&%#x3GF&F8F&F&*&F-F&F8F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,:*(\"\"'\"\"\")%#y2G\"\"#F&%#x3GF&!\"\"**F%F&%#y1GF&%# x1GF&%#y3GF&F&**F%F&%#x2GF&F-F&F/F&F+*(F)F&)F1F)F&%\"yGF&F+*(F%F&F*F&) F-F)F&F&*&)F*F)F&F4F&F+*&F-F&F4F&F&**F%F&F(F&F1F&F/F&F+**F)F&F1F&F*F&F 4F&F&*&F/F&F4F&F&**F%F&F.F&F(F&F/F&F&*&F(F&F4F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*(\"#=\"\"\"%#y2GF&)%#y1G\"\"#F&!\"\"*(F%F&F)F&)F'F*F &F+*&F%F&)F'\"\"$F&F&*$)%\"yGF*F&F&*&F%F&)F)F0F&F&*(\"#OF&F-F&%#y3GF&F &*(F7F&F(F&F8F&F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%#x1G\"\"#\" \"\"F(*$)%#x2GF'F(F(*&F'F(%#y1GF(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#x1G\"\"\"%#x2GF&F&%#y2G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.%#y1G\"\"\"*$)%#x3G\"\"#F%F%*&%#x1GF%F(F%F%*&%#x2GF%F(F%!\"\"% #y3GF.%#y2GF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "vars:=map(indets,GB);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%%varsG70<&%#x2G%#y1G%#y2G%#x1G<(%\"yG%#x3GF'F(F)F*< )F,F-F'F(F)%#y3GF*<(F-F'F(F)F/F*F.F.F.F.F.F.<&F,F(F)F/<%F'F(F*<%F'F)F* F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(print,vars);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<&%#x2G%#y1G%#y2G%#x1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(%\"yG%#x3G%#x2G%#y1G%#y2G%#x1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)%\"yG%#x3G%#x2G%#y1G%#y2G%#y3G%#x1G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<(%#x3G%#x2G%#y1G%#y2G%#y3G%#x1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)%\"yG%#x3G%#x2G%#y1G%#y2G%#y3G%#x1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)%\"yG%#x3G%#x2G%#y1G%#y2G%#y3G%#x1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)%\"yG%#x3G%#x2G%#y1G%#y2G%#y3G%#x1G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<)%\"yG%#x3G%#x2G%#y1G%#y2G%#y3G%#x1G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)%\"yG%#x3G%#x2G%#y1G%#y2G%#y3G%# x1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)%\"yG%#x3G%#x2G%#y1G%#y2G%#y3 G%#x1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&%\"yG%#y1G%#y2G%#y3G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%%#x2G%#y1G%#x1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%%#x2G%#y2G%#x1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<( %#x3G%#x2G%#y1G%#y2G%#y3G%#x1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "GB1:=gbasis([II1,II2,II 3],[x1,x2,x3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GB1G7(*$)%#x2G\" \"$\"\"\",&*&)F(\"\"#F*%#x3GF*F**&F(F*)F/F.F*!\"\"*$)F/F)F*,&*$)%#x1GF .F*F**$F-F*F**&F8F*F(F*,(*$F1F*F**&F(F*F/F*F2*&F8F*F/F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "# mkbasis(GB1,[x1,x2,x3],6); # Thi s procedure doesn't work -- missing procedure testmon" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rho:=REY(GG211,x3^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG,.*$)%#x3G\"\"#\"\"\"F**&#F*F)F**$)%#x1GF)F*F*F**&F/F*%#x 2GF*!\"\"*&F/F*F(F*F**&F,F**$)F1F)F*F*F**&F1F*F(F*F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "normalf(rho,GB1,[x1,x2,x3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rho2:=REY(GG211,x2*x3^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% rho2G,.*&#\"\"\"\"\"%F(*&)%#x1G\"\"#F(%#x2GF(F(F(*&F'F(*&F,F()F.F-F(F( F(*&#F(F)F(*$)F.\"\"$F(F(!\"\"*&#F(F-F(*&F1F(%#x3GF(F(F(*&#F(F)F(*$)F, F6F(F(F7*&#F(F-F(*&F+F(F;F(F(F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "factor(rho2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"%!\" \",&%#x1GF&%#x2G\"\"\"F*,&F)F*F(F*F*,(F)F&*&\"\"#F*%#x3GF*F*F(F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "J:=jacobian([II1,II2,II3] ,[x1,x2,x3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JGK%'matrixG6#7%7 %%#x1G%#x2G\"\"!7%F+F*F,7%,(F*\"\"\"F+!\"\"%#x3GF0,(F*F1F+F0F2F1,(F+F1 *&\"\"#F0F2F0F0F*F0Q*pprint5996\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "factor(det(J));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *(,&%#x1G!\"\"%#x2G\"\"\"F),&F(F)F&F)F),(F(F'*&\"\"#F)%#x3GF)F)F&F)F)F '" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 57 "The subgroup <(1,2),(3,4)> of S4, under the rep'n \+ A^(3,1)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "GG4:=[[1,2,3,4],[ 2,1,3,4],[1,2,4,3],[2,1,4,3]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$G G4G7&7&\"\"\"\"\"#\"\"$\"\"%7&F(F'F)F*7&F'F(F*F)7&F(F'F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "prep([3,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(Q*pprint60 06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "GG31:=map(NAT,GG4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%GG31G7&K%'matrixG6#7%7%\"\"\"\" \"!F,7%F,F+F,7%F,F,F+Q*pprint6016\"KF'6#7%7%!\"\"F5F5F-F.Q*pprint602F0 KF'6#7%F*F.F-Q*pprint603F0KF'6#7%F4F.F-Q*pprint604F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(jordan,GG31);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&K%'matrixG6#7%7%\"\"\"\"\"!F*7%F*F)F*7%F*F*F)Q*pprint 6056\"KF%6#7%7%!\"\"F*F*F+F,Q*pprint606F.KF%F0Q*pprint607F.KF%6#7%F2F+ 7%F*F*F3Q*pprint608F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "FR G31:=factor(hilb(GG31));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&FRG31G, $*&\"\"\"F'*&),&%\"qGF'F'!\"\"\"\"$F'),&F+F'F'F'\"\"#F'F,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "series(FRG31,q,20);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+M%\"qG\"\"\"\"\"!F%F%\"\"$\"\"#F'F'\" \"'\"\"%F)\"\"&\"#5F)F,\"\"(\"#:\"\")F.\"\"*\"#@F,F1\"#6\"#G\"#7F3\"#8 \"#O\"#9F6F.\"#X\"#;F8\"#<\"#b\"#=F;\"#>-%\"OG6#F%\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "II1:=REY(GG31,2*x3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$II1G,(%#x3G\"\"\"%#x1G!\"\"%#x2GF'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "II2:=REY(GG31,x1^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$II2G*$)%#x1G\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "II3:=REY(GG31,x2*x3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$II3G,**&%#x2G\"\"\"%#x3GF(F(*&#F(\"\"#F(*$)%# x1GF,F(F(F(*&#F(F,F(*&F/F(F)F(F(!\"\"*&#F(F,F(*&F/F(F'F(F(F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "J:=jacobian([II1,II2,II3],[x 1,x2,x3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JGK%'matrixG6#7%7%! \"\"\"\"\"F+7%,$*&\"\"#F+%#x1GF+F+\"\"!F17%,(F0F+*&F/F*%#x3GF+F**&F/F* %#x2GF+F*,&F5F+*&F/F*F0F+F*,&F7F+*&F/F*F0F+F*Q*pprint6096\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#x1GF&,&%#x2GF&%#x3G!\"\"F&F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "I6:=REY(GG31,(x2*x3)^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I6G,4*&)%#x2G\"\"#\"\"\")%#x3GF)F*F **&#F*F)F**$)%#x1G\"\"%F*F*F**&)F1\"\"$F*F,F*!\"\"*&F.F**&)F1F)F*F+F*F *F**&F4F*F(F*F6**F)F*F9F*F(F*F,F*F**(F1F*F(F*F+F*F6*&F.F**&F9F*F'F*F*F **(F1F*F'F*F,F*F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "GB:=gb asis([II1-y1,II2-y2,II3-y3,I6-y],[x1,x2,x3,y1,y2,y3,y]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#GBG7',6*&\"\"%\"\"\")%#x3G\"\"$F)F)*(\"\"'F)) F+\"\"#F)%#y1GF)!\"\"*&%#x2GF))F1F0F)F2*&F+F)F5F)F)*$)F1F,F)F)*(F(F)%# y2GF)F+F)F2*(F0F)F:F)F1F)F)*(F(F)F4F)%#y3GF)F)*(\"\")F)F+F)F=F)F)*(F.F )F1F)F=F)F2,(*&F:F)F5F)F)*&F(F))F=F0F)F)*&F(F)%\"yGF)F2,.*$)F4F0F)F)*$ F/F)F)*&F4F)F1F)F2*&F+F)F1F)F2F:F2*&F0F)F=F)F),,*(F0F)F4F)F+F)F)FKF2FL F2*$F5F)F)*&F0F)F=F)F2,*F+F2%#x1GF)F4F2F1F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "map(print,GB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,6*&\"\"%\"\"\")%#x3G\"\"$F&F&*(\"\"'F&)F(\"\"#F&%#y1GF&!\"\"*&%#x2GF& )F.F-F&F/*&F(F&F2F&F&*$)F.F)F&F&*(F%F&%#y2GF&F(F&F/*(F-F&F7F&F.F&F&*(F %F&F1F&%#y3GF&F&*(\"\")F&F(F&F:F&F&*(F+F&F.F&F:F&F/" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,(*&%#y2G\"\"\")%#y1G\"\"#F&F&*&\"\"%F&)%#y3GF)F&F&*& F+F&%\"yGF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%#x2G\"\"#\" \"\"F(*$)%#x3GF'F(F(*&F&F(%#y1GF(!\"\"*&F+F(F-F(F.%#y2GF.*&F'F(%#y3GF( F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(\"\"#\"\"\"%#x2GF&%#x3GF&F&* &F'F&%#y1GF&!\"\"*&F(F&F*F&F+*$)F*F%F&F&*&F%F&%#y3GF&F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,*%#x3G!\"\"%#x1G\"\"\"%#x2GF%%#y1GF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "NGB:=gbasis([I1,I2,I3],[x1,x2,x3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$NGBG7#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "# mkbasis(NGB,[x1,x2,x3],6); # This doesn't work, missing pro cedure testmon" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "P:=(1+2*q+q^2)/(1-q)^2/(1-q^ 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG*(,(\"\"\"F'*&\"\"#F'%\"q GF'F'*$)F*F)F'F'F',&F'F'F*!\"\"!\"#,&F'F'F+F.F." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "factor(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*&,&%\"qG\"\"\"F'F'F',&F&F'F'!\"\"!\"$F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "S3 x S2" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "S32:=conca(permute([1,2,3]) ,permute([4,5]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S32G7.7'\"\"\" \"\"#\"\"$\"\"%\"\"&7'F'F(F)F+F*7'F'F)F(F*F+7'F'F)F(F+F*7'F(F'F)F*F+7' F(F'F)F+F*7'F(F)F'F*F+7'F(F)F'F+F*7'F)F'F(F*F+7'F)F'F(F+F*7'F)F(F'F*F+ 7'F)F(F'F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "SS32:=map(m kpermat,S32):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "FRS32:=hil b(SS32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&FRS32G,,*&\"\"\"F'*&\"# 7F',.F'F'*&\"\"&F'%\"qGF'!\"\"*&\"#5F')F-\"\"#F'F'*&F0F')F-\"\"$F'F.*& F,F')F-\"\"%F'F'*$)F-F,F'F.F'F.F'*&F'F'*&F5F',.F'F'*&F2F'F1F'F'*&F5F'F 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F'F'F9F.*&F2F'F7F'F'*$F4F'F.F'F.F'*&F'F'*&FJF',*F9F'FOF.F'F'FLF.F'F.F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(FRS32);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*(),&%\"qGF%F%F%\"\"#F%,(* $)F)F*F%F%F)F%F%F%F%),&F)F%F%!\"\"\"\"&F%F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "series(FRS32,q,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"qG\"\"\"\"\"!\"\"#F%\"\"&F'\"\"*\"\"$\"#;\"\"%\"#D F(\"#R\"\"'\"#c\"\"(\"#!)\"\")\"$4\"F)-%\"OG6#F%\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rho:=REY(SS32,(x1+x2-x5)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG,>**\"\"#\"\"\"\"\"$!\"\"%#x1GF(%#x2GF(F (*&F'F*%#x4GF'F(**F'F(F)F*%#x3GF(%#x5GF(F***F'F(F)F*F0F(F.F(F**(F'F(F) F*F0F'F(**F'F(F)F*F+F(F0F(F(*(F'F(F)F*F,F'F(*(F'F(F)F*F+F'F(**F'F(F)F* F,F(F0F(F(*&F'F*F1F'F(**F'F(F)F*F+F(F1F(F***F'F(F)F*F,F(F1F(F***F'F(F) F*F+F(F.F(F***F'F(F)F*F,F(F.F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sort(factor(rho));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,>*(\"\"#\"\"\"\"\"$!\"\"%#x3GF%F&**F%F&F'F(F)F&%#x2GF&F&**F%F&F'F(F) F&%#x1GF&F&**F%F&F'F(F)F&%#x4GF&F(**F%F&F'F(F)F&%#x5GF&F(*(F%F&F'F(F+F %F&**F%F&F'F(F+F&F-F&F&**F%F&F'F(F+F&F/F&F(**F%F&F'F(F+F&F1F&F(*(F%F&F 'F(F-F%F&**F%F&F'F(F-F&F/F&F(**F%F&F'F(F-F&F1F&F(*&F%F(F/F%F&*&F%F(F1F %F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "REY(SS32,rho);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,>*(\"\"#\"\"\"\"\"$!\"\"%#x3GF%F&**F% F&F'F(F)F&%#x2GF&F&**F%F&F'F(F)F&%#x1GF&F&**F%F&F'F(F)F&%#x4GF&F(**F%F &F'F(F)F&%#x5GF&F(*(F%F&F'F(F+F%F&**F%F&F'F(F+F&F-F&F&**F%F&F'F(F+F&F/ F&F(**F%F&F'F(F+F&F1F&F(*(F%F&F'F(F-F%F&**F%F&F'F(F-F&F/F&F(**F%F&F'F( F-F&F1F&F(*&F%F(F/F%F&*&F%F(F1F%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 15 } {VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }