Instructor: Brendon Rhoades
Instructor's Email: bprhoades (at) math.ucsd.edu
Instructor's Office: 7250 APM
Instructor's Office Hours: 4:00-5:00 pm MWF
Lecture Time: 3:00-3:50pm MWF
Lecture Room: 5402 APM
Final Exam Time: Wednesday, 3/22/2017, 3:00pm-5:59pm
Final Exam Room: TBA

Syllabus: Coming Soon!

Lectures:

1/9: Administrivia. Section 1.1. Permutations. Two-line, one-line, cycle notation. Conjugacy in groups. Cycle notation and conjugacy. Partitions. Formula for size of conjugacy classes in S_n. Centralizers. The sign of a permutation.
1/11: Section 1.2. Matrix representations of groups G over fields k. Degree. Examples. Section 1.3. G-modules. Examples. G-modules from actions of G on sets. The group algebra k[G]. G-submodules. The equivalence of G-modules and matrix representations.
1/13: Coset representations k[G/H]. Irreducibility. G-module homomorphisms, endomorphisms, and isomorphisms. The direct sum of G-modules. Indecomposability. Statement of Maschke's Theorem.
1/18: Proof of Maschke's Theorem. Section 1.6. Schur's Lemma. Examples and non-examples. Section 1.7. Commutant algebra Com_X of a matrix representation X.
1/20: More on commutant algebras. The center Z_A of an algebra A. Formulas for the dimension of Com_X and its center (if k is algebraically closed). Section 1.8. Characters. Class functions. Character tables. Examples: C_4 and S_3.
1/23: Section 1.9. Character inner product. Class function inner product. Character orthogonality of the first kind. Corollaries. Section 1.10. Decomposition of the regular representation C[G]. Magic formula.
1/25: The number of irreps of a finite group G over C equals the number of conjugacy classes of G. Character orthogonality relations of the second kind. Definition of the tensor product of k-vector spaces V and W.
1/27: If V is a G-module and W is an H-module, then V tensor W is a (GxH)-module. Irreps of direct products GxH. 1/30: Restriction of G-modules to subgroups H. Induction of H-modules to supergroups G. Induction of characters. Induced trivial representations are coset representations. Frobenius reciprocity.
2/1: The character table of D_n for n odd. Young subgroups. Tableaux and tabloids. The tabloid module M^{lambda}. Posets. Dominance order on partitions.
2/3: The Dominance Lemma. Row and column stabilizers. Polytabloids. Specht modules S^{lambda}. Preview of results on Specht modules.
2/6: Proof that the Specht modules constitute a complete set of S_n-irreps. Decomposition of the tabloid module M^{mu} into the S^{lambda}'s.
Semistandard tableaux and statement of Young's Rule.
2/8: Proof that standard polytabloids form a basis for Specht modules (dominance order on tabloids, column tabloids, Garnir elements). Magic formula for S_n.
2/10: Young's Natural Representation. The Branching Rule for symmetric groups. Gelfand-Tzetlin bases.
2/13: Semistandard tableaux. Shape and content. Young's Rule. Sketch of proof. Section 3.1. The Schensted correspondence: permutations, words, N-matrices.
2/15: Section 4.1. Generating functions. Euler's product formula. "Odd = distinct" theorem. Section 4.3. Monomial symmetric functions. The ring of symmetric functions. Power sum, elementary, and homogeneous symmetric functions.
2/17: Section 4.3. The e_{lambda}, p_{lambda}, and m_{lambda} symmetric functions. The e's, p's, and h's form bases for the ring of symmetric functions. Section 4.4. Schur functions (SSYT definition). The Schur functions are symmetric: Bender-Knuth involutions.
2/22: Section 4.4. The Schur functions are a basis for the ring of symmetric functions. Section 4.5. The Jacobi-Trudi formula. Lindstrom-Gessel-Viennot theory.
2/24: Section 4.6. Statement of the bialternant formula for Schur polynomials. The transition matrix from the p-basis to the m-basis. The transition matrix from the s-basis to the p-basis. Section 4.7. The Frobenius character map. The Hall inner product on symmetric functions. The Frobenius character map ch is an isometry.
2/27: Section 4.7. More examples of Frobenius characters. Induction product of symmetric group class functions. The Frobenius character map is a ring isomorphism. ch(S^{lambda}) = s_{lambda}, ch(M^{lambda}) = h_{lambda}. A module N^{lambda} with character e_{lambda}. A class function psi^{lambda} with character p_{lambda}.
3/1: Monk's Rule/Branching Theorem for symmetric groups. Statement of Pieri and dual Pieri rule. Littlewood-Richardson tableaux. Littlewood-Richardson Rule.
3/3: Rim hooks in partitions. Statement of the Murnaghan-Nakayama Rule. Example. Skew shapes. Skew tableaux. Skew Schur functions. Cauchy's Identity.
3/6: Skew shape version of the LR rule. Sketch of skew shape version (assuming the Fundamental Theorem of jeu-de-taquin).
3/8: No class; Prof. Rhoades was in Pasadena.
3/10: Proof of Murnaghan-Nakayama rule using the Littlewood-Richardson rule.
3/13: The hook length formula. Probabilistic proof using hook walks by Greene-Nijenhuis-Wilf.
3/14: Make-up Lecture. The representation theory of GL(V), for V an n-dimensional C-vector space. Rational and polynomial representations. Weyl characters. Schur functors. Schur-Weyl duality. Decomposition of tensor space V x ... x V using RSK.
3/15: Symmetric function wrap-up. The omega map. Multiplication and co-multiplication. The Hopf algebra of symmetric functions.
3/17: Plethysm of symmetric functions. Plethysm and Schur functors. Review.

Homework Assignments:

Homework 1, due 1/18/2017.
Homework 2, due 1/25/2017.
Homework 3, due 2/1/2017.
Homework 4, due 2/15/2017.
Homework 5, due 2/24/2017.
Homework 6, due 3/13/2017.