Winter 2018

**Instructor:** Brendon Rhoades

**Instructor's Email:** bprhoades (at) math.ucsd.edu

**Instructor's Office:** 7250 APM

**Instructor's Office Hours:** By appointment (after class works well)

**Lecture Time:** 1:00-1:50pm MWF

**Lecture Room:** 5402 APM

**TA:** Dun Qiu

**TA's Email:** duqiu (at) ucsd.edu

**TA's Office:** E106 SDSC

**Final Exam Time:** Friday, 3/23/2018, 11:30am-2:29pm

**Final Exam Room:** 5292 APM

** Lectures:**

**1/8:** Section 1.1. The symmetric group S_n. One-line, two-line, cycle notation
for permutations. Cycle type. Partitions. Number of permutations with a given cycle type.
Conjugacy classes and centralizers. Conjugacy in S_n. The sign of a permutation.
Section 1.2. Matrix representations of groups. Examples. Permutation matrices.

**1/10:** Section 1.2. More examples of matrix representations. Degree 1 representations
of C_n over the complex numbers. Section 1.3. G-modules. Examples. Actions of groups on
sets; the permutation representaions k[S]. The group algebra. The regular representation.
Coset representations k[G/H]. Equivalence of finite-dimensional G-modules and matrix
representations. Section 1.4. Submodules. Examples. Irreducibility.

**1/12:** Section 1.4. Submodules in terms of matrices.
G-module homomorphisms. G-module homomorphisms/isomorphisms in terms of matrices.
Kernel and image.
Section 1.5. Module direct sum. Indecomposability. Complete reducibility.
Maschke's Theorem.

**1/17:** Section 1.5. Matrix version of Maschke's Theorem. Section 1.6.
Baby Schur's Lemma. Real Schur's Lemma. Failure of real Schur over R.
Section 1.7. End(V) and Com(X). Matrix tensor product. Description of Com(X)
as an algebra of matrices.

**1/19:** Section 1.8. Centers of algebras. Dimension of the center of Com(X).
Tensor products of vector spaces. Operation within tensor products. Universal
property of tensor products. Bases of tensor products. Induced linear maps between
tensor products; connection to tensor products of matrices. If V is a G-module
and W is an H-module, the G x H structure on V tensor W. Tensor products of
matrix representations.

**1/22:** More multilinear constructions: symmetric and exterior powers.
Section 1.8. Characters of representations/modules. Examples. Class functions.

**1/24:** G-invariant inner products. Unitary representations.
Section 1.9. Inner product on class functions. Character Orthogonality of the First
Kind. Corollaries: two G-modules V and W are isomorphic iff they have the same
character. Formula for a character inner product with itself.
Section 1.10. Decomposition of the group algebra C[G] into irreducible representations.

**1/26:** More on C[G]. The number of irreducible characters of G equals
the number of conjugacy classes of G. Character tables. Character orthogonality
of the second kind. Character tables of S_3, C_4, S_4.

**1/29:** Section 1.11. Representations of product groups G x H via tensor products.
Section 1.12.
Induction and restriction of modules. Induction and restriction of characters.
Induction via tensor products.

**1/31:** The coset representation C[G/H] as an induced trivial representation.
Frobenius reciprocity on class functions. The character table of D_7.
Section 2.1. Partitions, tableaux, and tabloids. The tabloid module
M^{lambda}.

**2/2:** Section 2.1. Young subgroups. The M^{lambda} as induced modules. Section 2.2.
Dominance order on partitions. The Dominance Lemma. Section 2.3. The row
and column groups R_T and C_T of a tableau T.
Young antisymmetrizers kappa_T and polytabloids e_T. Definition of
the modules S^{lambda}. Proof that S^{lambda} is closed under the action of S_n.
Examples of S^{lambda} when lambda = (n) or (1^n).

**2/5:** Section 2.4. Inner product on M^{lambda}. Sign lemma.
Antisymmetrizer version of Dominance Lemma. Submodule Theorem.
The S^{lambda} are irreducible. If lambda and mu are different partitions of n,
S^{lambda} and S^{mu} are non-isomorphic. Decomposition of M^{mu} into
S^{lambda}'s. Section 2.5. Standard Young tableaux.

**2/7:** No class. Prof. Rhoades is in LA.

**2/9:** Section 2.5. Compositions. Dominance order on compositions.
Dominance order on tabloids. Proof that the standard polytabloids are
linearly independent. Section 2.6. Garnir elements.

**2/12:** Section 2.6. Proof that the standard polytabloids form
a basis for S^{lambda}. Section 2.7.
Young's Natural Representation.
Section 2.8. Branching rule for symmetric groups.
Branching rules for other towers of algebraic objects
(e.g. Temperley-Lieb algebras).

**2/14:** Section 2.8. Proof of the branching rule for
symmetric groups. Section 2.9-11. Semistandard tableaux:
shape/content. Statement of Young's Rule. Special cases.
**2/16:** Section 4.1. Ordinary generating functions.
Euler's formula. Odd = distinct theorem.
Section 4.3. The monomial symmetric functions. The ring
of symmetric functions. Power sum, elementary, and homogeneous
symmetric functions.

**2/21:** Section 4.3. Power sum, monomial, elementary, and
homogeneous symmetric functions form bases for the ring of
symmetric functions. Section 4.4. Combinatorial definition of Schur
functions. Schur functions are symmetric. Schur functions form
a basis of symmetric functions. Section 4.5. Jacobi-Trudi identity
statement.

**2/23:** Section 4.5. Planar networks.
Path matrices. Lindstrom-Gessel-Viennot. Proof of Jacobi-Trudi
using planar networks. Section 4.6. Bialternant formula
for Schur polynomials. Power sum to monomial transition matrix
via phi^{mu} characters. Schur function to power sum transition
matrix using irreducible chi^{lambda} characters.

**2/26:** Section 4.7. The inner product space R_n.
The Frobenius character ch_n: R_n -> Lambda_n. Hall inner product
on symmetric functions. ch_n is an isometry. The space
R = (direct sum over n) R_n. Induction product on characters/modules.
ch_n is an algebra isomorphism. M^{lambda} as an induction product.
Section 4.8. Intro to RSK.

**2/28:** The RSK correspondence on permutations:
the magic formula for S_n. The RSK correspondence on words:
decomposition of the regular representation of S_n.
The RSK correspondence on N-matrices: the Cauchy Identity.

**3/2:** Section 4.8. Monk's Rule. Pieri Rule.
Dual Pieri Rule. Skew shapes and skew tableaux.
Ballot and reverse ballot sequences. Statement of the
Littlewood-Richardson Rule. Application to cohomology of
Grassmannians.

**3/5:** LR Rule in terms of skew shapes. Jeu-de-taquin. Sketch of proof
of LR Rule using jeu-de-taquin. Rim hooks and height. Statement of the
Murnaghan-Nakayama Rule. Examples.

**3/7:** Proof of Murnaghan-Nakayama Rule.

**3/9:** Conjugacy and tensoring with the sign representation.
The omega involution on symmetric functions. Hook lengths.
The Hook Length Formula; proof using the Hook Walk.

**3/12:** The character table of S_5.
Intro to the representation theory of GL_n(C).
Representations from field automorphisms.
Failure of semi-simplicity. Polynomial representations.
Tensor, exterior, and symmetric powers. Characters and
Weyl characters.

**3/14:**
Allusion to the Weyl Unitary Trick.
Calculation of Weyl characters. Weyl characters
are symmetric polynomials. Behavior under direct sum and tensor
product. Young idempotents. Schur functors and Schur polynomials.
Decomposition of tensor space into irreducibles.

**3/15:**
Bonus lecture! Interesting bases for S_n-irreducibles.
Noncrossing perfect matchings. Action via Coxeter presentation.
Connection to cyclic sieving.
Connection to Gr(2,n).

** Homework Assignments: **

Homework 1, due 1/19/2018.

Homework 2, due 2/2/2018.

Homework 3, due 2/16/2018.

Homework 4, due 3/2/2018.