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.

** Homework Assignments: **

Homework 1, due 1/19/2018.

Homework 2, due 2/2/2018.

Homework 3, due 2/16/2018.