Winter 2015

**Instructor:** Brendon Rhoades

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

**Instructor's Office:** 7250 APM

**Instructor's Office Hours:** 10:00-11:00am MWF or by appointment

**Lecture Time:** 2:00-2:50pm MWF

**Lecture Room:** 5402 APM

**Final Exam Time:** Monday, 3/16/2015, 3:00pm - 5:59pm

**Final Exam Room:** TBA

** Grading:**

Homework: 60%

Final Exam: 40%

** Textbook:** *The Symmetric Group* by Bruce Sagan

** Lectures:**

**1/5:** Section 1.1. Permutations. Two-line notation.
One-line notation. Cycle notation. Multiplication convention.
Conjugacy classes and centralizers. Cycle type.
Two permutations are conjugate if and only if they have the same
cycle type.

**1/7:** Section 1.1. The sizes of the conjugacy class and centralizer
of a permutation in S_n. The sign of a permutation.
Section 1.2. Group homomorphisms. Mat_d(k) and GL_d(k) for a field k.
Matrix representations of groups. The degree of a representation.
Examples: The trivial representation.
Degree 1 representations of finite cyclic groups.
The sign representation of the symmetric group.

**1/9:**
Section 1.2. Defining representation of S_n.
Section 1.3. GL(V) and G-modules. Relation with matrix
representations. Example: Dihedral group. The vector space k[S]
for a set S and the group algebra k[G]. The left regular representation
of a group G.

**1/12:**
Section 1.4. G-submodules of a G-module V. Irreducibility.
Section 1.5. Complements. G-invariant inner products.
Maschke's Theorem. Complete reducibility.

**1/14:**
Section 1.6. G-homomorphisms and G-isomorphisms.
Schur's Lemma. Section 1.7. The endomorphism algebra
End(V) = End_G(V) and the commutant algebra
Com(X) = Com_G(X) of a matrix representation X.
The structure of Com(X) (over an algebraically closed field) for
X = m_1 X^1 + ... + m_r X^r
for X^1, ... , X^r pairwise nonisomorphic irreducibles.

**1/16:** Section 1.7. The center Z_A of an algebra A. The center of
End(V) for a G-module V. Section 1.8. The character of a representation.
Class functions. Character tables (example: S_3).
Section 1.9. Character
inner product. Statement of Character Orthogonality of the
First Kind.

**1/21:** Section 1.9. Proof of Character Orthogonality of the
First Kind. Representations are determined up to isomorphism by their
characters. If X has character chi, then X is irreducible iff
< chi, chi > = 1. Section 1.10. Decomposition of the group algebra C[G].
Magic formula. The number of irreducible characters equals the number
of conjugacy classes. The characters give an orthonormal basis for
the space of class functions.

**1/23:** Review of vector space direct sum.
Definition of vector space tensor product as a quotient space.
Tensor space universal property.
Bases of tensor spaces.
Linear maps between tensor spaces and relationship
with matrix tensor product.

**1/26:** Section 1.11. Tensor products of representations.
Irreps of G x H. External direct sum and Kronecker product
of representations. Section 1.12. Definition of restricted
and induced representations.

**1/28:** Section 1.12. Induced representations are actually
representations and independent (up to isomorphism) of
the choice of transversal. Induced characters.
Frobenius reciprocity. Section 2.1.
Ferrers diagrams of partitions and Young subgroups.

**1/30:** Section 2.1. Tableaux and tabloids.
The tabloid modules M^{lambda}.
Section 2.2. Posets, dominance order, and lexicographical order.
Section 2.3. Row stabilizers, column stabilizers, and polytabloids.
Definition of the Young modules.

**2/2:** Section 2.4. The sign lemma. The Young modules
are a complete set of pairwise nonisomorphic irreducible
representations of S_n.

**2/4:** Section 2.5. Standard (Young) tableaux. Big Theorem:
The standard polytabloids form a basis for the Young modules.
Dominance order on partitions and tabloids. Proof that
the standard polytabloids are linearly independent.
Section 2.6.
Definition of Garnir elements.

**2/6:** Section 2.6. More on Garnir elements.
Column tabloids. Proof that standard polytabloids span Young
modules. Section 2.7. Young's Natural Representation.
Section 2.8. Statement of the Branching Rule for symmetric groups.

**2/9:** Section 2.8. Proof of the Branching Rule for
symmetric groups. Section 2.9-11. Generalized tableaux.
Shape and content. Semistandard tableaux and Kostka numbers.
Young's Rule.

**2/11:** Section 2.9-11. Young's Rule wrap up, idea of proof.
Section 4.1. Ordinary generating functions. Generating function
for partitions. "Odd = distinct". Convergence
of infinite products of formal power series.

**2/13:** Section 4.3. The ring of symmetric functions.
Monomial, power sum, elementary, and complete homogeneous symmetric functions.
These sets are bases of the ring of symmetric functions.

**2/20:** Section 4.4. Combinatorial definition of the Schur
functions. Bender-Knuth involution and proof the Schur functions are
symmetric. Schur functions are a basis for the ring of symmetric functions.
Section 4.5. Statement of the Jacobi-Trudi formula.
Lindstrom/Gessel-Viennot theory.

**2/23:** Section 4.5. Proof of the Jacobi-Trudi formula.
Section 4.6. Statement of the bialternant formula for Schur functions
(in finitely many variables).
The transition matrix from the power sum to the monomial
symmetric functions is given by the tabloid/Young coset characters.

**2/25:** Section 4.6. The transition matrix from the Schur basis
to the power sum basis is (up to an easy scalar)
the character table of S_n.
Section 4.7. The characteristic map (Frobenius character).
Hall inner product. The characteristic map is an isometry
{class functions on S_n} ---> {symmetric functions of degree n}.
``Bilinear form" formula for ch(f), f a class function.
ch(V) for an S_n-module V.
Two S_n-modules are isomorphic iff they have the same Frobenius character.
The multiplicity of S^{lambda} in V is the coefficient of
s_{lambda} in the Schur basis expansion of ch(V).
Induction product on modules/characters.
Under induction product on R = {all class functions symmetric groups --> C},
ch: R ---> (ring of symmetric functions)
is an isomorphism of rings.

**2/27:** Section 4.7. More on induction products.
Section 4.8. The Frobenius character of tabloid representations.
The transition matrix from complete homogeneous symmetric functions
to Schur functions. Statement of the hook length formula.

**3/2:** Review material on the characteristic map.
Ribbons and rim hooks. Statement of the Murnaghan-Nakayama
Rule and an example.

**3/4:** Monk's Formula. The Pieri Rule. The Pieri Rule
implies Young's Formula. Ballot and reverse ballot sequences.
Skew tableaux and row words. Statement of the Littlewood-Richardson
Rule.

**3/6:** The Littlewood-Richardson rule: Examples. The Dual
Pieri Rule. The omega involution. Section 3.1. The RSK correspondence
on permutations. Theorems of Schensted and Schutzenberger.

**3/9:** The RSK correspondence on words. Relationship to
decomposition of the regular representation of S_n. Section 4.8.
The RSK correspondence on biwords/N-matrices.
Generalization of Schutzenberger's Theorem.
The Cauchy identity.

** Homework Assignments: **

Homework 1, due 1/16/2015.

Homework 2, due 1/23/2015.

Homework 3, due 1/30/2015.

Homework 4, due 2/6/2015.

Homework 5, due 2/13/2015.

Homework 6, due 2/27/2015.

Homework 7, due 3/11/2015.