Instructor: Brendon Rhoades
Instructor's Email: bprhoades (at) math.ucsd.edu
Instructor's Office: 7250 APM
Instructor's Office Hours: By appointment
Lecture Time: 11:00-11:50pm MWF
Lecture Room: 5402 APM
Grading:
Homework: 100%
Textbook: The Symmetric Group by Bruce Sagan
Lectures:
1/7: Administrivia.
Review of basic group theory.
Basic facts about S_n.
1/9: Matrix representations of groups (over arbitrary fields K).
Degree. Defining/sign representations of S_n. Trivial representation.
G-modules. Examples. G-module direct sum. Permutation representations.
The regular representation K[G].
1/11: Coset representations K[G/H]. Module direct sum.
Submodules. Irreducibility. G-homomorphisms.
1/14: Indecomposability vs irreducibility. Maschke's Theorem and complete
reducibility.
1/16: Baby Schur's Lemma. True Schur's Lemma. The structure of
End_G(V) and Hom_G(V,W) in terms of matrices (matrix tensor product).
1/18: Characters and class functions. Unitary matrices and
G-invariant inner products. Inner product on R(G).
1/28: Character orthogonality of the first kind.
Representations are determined by their characters. Decomposition of C[G].
1/30: Centers of algebras. The number of irreps of G is the number of conjugacy
classes. Character tables. Examples: C_4, D_4, S_4.
2/1: Tensor product of matrix representations. Representations of G x H.
Tensor product of vector spaces. Universal property.
2/4: Module Tensor Product. Induction and restriction.
2/6:
Frobenius Reciprocity. Character table of D_7.
2/8: Partitions, tableaux, and tabloids. Young subgroups.
The tabloid representations M^{lambda}. Dominance order and the
Dominance Lemma.
2/11:
The Specht modules are a complete set of
nonisomorphic S_n-irreps. Proof using
Submodule Theorem. Intro to the decomposition of M^{mu}.
2/13:
Standard polytabloids form a basis for Specht modules.
Linear independence: dominance on tabloids. Spanning: Garnir elements.
2/15:
Branching rule.
Semistandard tableaux. Kostka numbers. Statement of
Young's Rule.
2/20: Sketch of proof of Young's Rule.
Formal power series. Ordinary generation functions.
Partition identities. Odd = distinct.
2/22:The ring of symmetric functions. Monomial, homogeneous, elementary,
power sum symmetric functions. The omega involution. Combinatorial definition
of Schur functions.
2/25: Combinatorial proof of Schur symmetry: Bender-Knuth involutions.
Planar networks, Lindstrom-Gessel-Viennot theory and the Jacobi-Trudi
identity.
2/27:Alternants and the Bialternant Formula for Schur polynomials.
p-to-m transition and representation theory. s-to-p transition and representation theory.
Frobenius characteristic map. Hall inner product.
3/1: More on Frobenius images. Induction product. Examples of class functions
and their Frobenius images. Schensted correspondence on permutations.
Lecture Notes:
Lectures 1-10
Homework Assignments:
Homework 1, due 1/28.
Homework 2, due 2/15.
Homework 3, due 3/1.
Homework 4, due 3/15.