Syllabus
Office hours: MWF 10-11 am (right after class) in 873 Evans

Notes


Days 1-11,16 follow Pavel Etingof's notes.

Days 12-15 (the induction theorems) follow Curtis and Reiner's "Methods of Representation theory".

Days 17-21 follow Macdonald's "Symmetric functions and Hall polynomials".

Days 25-30 follow Fulton's and Harris's "Representation theory, a first course".

Days 31-37 follow the appendices to Fulton and Harris, or Humphreys's "Introduction to Lie algebras and Representation theory"

Days 38-40 follow Humphreys's "Representations of Semisimple Lie Algebras in the BGG Category O", chapters 1 and 2.


Day 1: Welcome to Representation theory!

Day 2: Finite dimensional associative algebras

Day 3: Semisimplicity and characters

Day 4-6: Guest lectures by Allen Knutson!

Day 7: Irreps of the symmetric group I

Day 8: Irreps of the symmetric group II

Day 9: Irreps of the symmetric group III

Day 10: Schur-Weyl duality

Day 11: Integrality and consequences

Day 12: Artin's induction theorem

Day 13: In class exercises on induction.

Day 14: Solomon's induction theorem

Day 15: Brauer's induction theorem

Day 16: Representations of GL2(Fq)

Day 17: Representations of GLn(Fq), I: conjugacy classes and parabolic induction

Day 18: Representations of GLn(Fq), II: Induction and the Hall algebra

Day 19: Hall algebra examples, and continuation of day 18 (see yesterday's notes).

Day 20: A crash course on symmetric functions

Day 21: Representations of GLn(Fq), III: Characters (this lecture leaves the proofs to Homework 3.)

Day 22: Counting covers of curves.

Day 23: Bundles on curves, 1.

Day 24: Bundles on curves, 2.

Day 25: Definition of a Lie algebra; SL2 (C)

Day 26: Finite dimensional representations of SL2(C)

Day 27: SL3(C), part I

Day 28: Examples of SL3 representations and weight diagrams

Day 29: Irreps of SL3 and their characters. (Day 28 and 29 involved too many pictures for making notes, but weren't too different from the chapters in Fulton and Harris.)

Day 30: Semisimple Lie Algebras; overview.

Day 31: Upper triangularity -- the theorems of Lie and Engel

Day 32: The Killing form and Cartan's criterion

Day 33: Semisimplicity

Day 34: The absolute Jordan decomposition

Day 35: Cartan subalgebras

Day 36: Root systems

Day 37: The Weyl group

Day 38: Highest weight modules; category O

Day 39: Classification of finite dimensional simples; central characters

Day 40: Harish-Chandra's theorem and the Weyl character formula

Homeworks

Homework 0: due Monday, Sept. 1; see Lecture 1

Homework 1: due Monday, Sept. 17

Homework 2: due Monday, Oct. 7

Homework 3: (Optional, more here later.) Read in Macdonald's book the rest of the proof that the characters of GLn(Fq) that we constructed really are the irreducible characters.

Homework 4: due Monday, Nov. 4; do all the exercises in Lecture 26.

Homework 5: due Friday, Nov. 15; do all the exercises in Lecture 30.

Homework 6: due Wednseday, Nov. 27; do all the exercises in Lectures 31, 32, 33, 34 (there are 5 total).

Some things that might have been on the final.