ICLUE Representation theory and Combinatorics program Summer 2022 (May 15-July 15, 2022)

Locations: Altgeld Common room, Noyes 163, Altgeld 245, Altgeld 314,...

Participants:

(Judy Chiang, Zhuo Zhang, Casey Appleton, AY, Dylan Roscow, Dinglong Wang, Dylan Chiu)

Some references:

B. Mazur's notes on equality, categories

A short primer on Young tableaux and symmetric functions

J.-P. Serre's finite group rep theory book

K. Smith's notes on rep theory

P. Etingof's notes on rep theory

C. Teleman's notes on rep theory

B. Conrad's essay on "Why study representation theory?"

J. Kamnitzer's notes on rep theory of compact groups and complex reductive groups

GL_n(C) for combinatorialists by Richard Stanley

Specht modules and Specht matroids by John D. Wiltshire-Gordon, Alexander Woo, Magdalena Zajaczkowska (Alexander Woo gave a personal lecture in Summer 2022 on this paper).

From Littlewood-Richardson coefficients to cluster algebras in three lectures by Andrei Zelevinsky

James Humphreys' Reflection groups and Coxeter groups book

Fulton and Harris' book on Representation theory, a first course

Instructor *raw* notes (text files, draft):

Introduction

Finite group characters

Things to know about the tensor product

The Frobenius character map and connection of S_n rep theory to symmetric functions (hand written notes to be posted). In this file: Young symmetrizers, the Specht modules (irreps of S_n), the Weyl construction (Schur functors) for GL_d irreps. Schur-Weyl duality.

Reflection groups, root systems, complex semisimple Lie algebras

Scribe notes:

Groups, Categories, Repesentations, by Judy Chiang and Dylan Roscow

Group rings, modules, examples, equivalence of reps, by Casey Appleton and Dylan Roscow

Polynomial and rational reps, permutations reps, Maschke's theorem, by Casey Appleton and Zhuo Zhang

Reps of (Z,+), dual reps, first steps into character theory, by Dylan Chiu and Zhuo Zhang

Schur's Lemma, Class functions and characters, Orthogonality, Character Tables, by Dylan Chiu and Dinglong Wang

Intersection between Rep theory and Algebraic geometry, the regular representation and its character, irreducible characters form a basis of the Class functions, relationship to Standard Young tableaux, onward to the tensor product, by Judy Chiang and Dinglong Wang

Invariant theory, the tensor product, by Judy Chiang and Dylan Roscow

More on tensor product, representation ring, group product, Frobenius character map, by Dylan Roscow and Zhuo Zhang

Semidirect product, matrix multiplication, algebra and lie algebra, tensor algebra, by Dylan Chiu and Zhuo Zhang

Graded Rings, Modules, and Algebras; The Symmetric Algebra; The Exterior Algebra; by Casey Appleton and Dylan Chiu

Alexander Woo's lecture on a construction of the Specht module; by Casey Appleton, Judy Chiang, and Dinglong Wang

Tensor square decomposition, Extension of scalars and induced representation, More category theory; by Dylan Roscow and Zhuo Zhang

Induced Representations, existence and uniqueness of induced representation, Induced characters, Induction via tensor product; by Zhuo Zhang

Littlewood-Richardson coefficients; by Dylan Roscow and Zhuo Zhang

Presentations

Hecke algebras and Kazhdan-Lusztig polynomials; by Zhuo Zhang

Lie groups and Haar Measure; by Dylan Roscow