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.