Winter 2017

**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.