## ** MATH603 (Spring 2022): Introductory topics in Representation theory.**

**Instructor: **Prof. Ivan Loseu (email: ivan.loseu@yale.edu)

**Lectures: **TTh 9-10:15am, Location: LOM 215 (the first two weeks are on zoom).
The first class is on **1/25**, and the last class is on **4/28**.

**Office hours: ** T 10.30-11.30am and W3-4pm and by appointment, DL 416.

The class discusses several topics in Representation theory including those of current interest.
This document contains
important information on the class content, prerequisites, homeworks, references, etc. Please read it!
More references will be posted on this page as the class progresses.

**Grading: **Based on around 4 homework sets (75%) and a final project or take-home final exam (25%).

**Homeworks:**

Homework 1 (updated on Feb 3 with hints and clarification and further updated on Feb 12), due **Feb 18**, by the end of day.

Homework 2, due **Mar 15**, by the end of day. Update (3/6): minus sign for F in problem 2.

Homework 3, due **Apr 7**, by the end of day. Update (3/27): definition of \Delta in the discussion before 3) in
problem 3 added.

Homework 4 (a.k.a. the last), due **Apr 29**, by the end of day.

A brief (and very condensed) write-up on the basics of Representation theory
(updated 1/8). For a more detailed treatment, please see the references for the course.

Notes on representations of symmetric groups, a.k.a. [RT1], variant of 2/3, possibly final.

**Schedule**:

Jan 25, Lecture 1: *Representations of symmetric groups following Okounkov and Vershik. Part 1 -- introduction and
centralizer algebras*. Notes from the lecture, covering Sections 1 and 2.1 in [RT1].

Jan 27, Lecture 2: *Representations of symmetric groups following Okounkov and Vershik. Part 2 -- centralizer algebras,
continued. Branching graph*. Notes from the lecture, covering Sections 2.2 and 3.1 in [RT1].

Feb 1, Lecture 3: *Representations of symmetric groups following Okounkov and Vershik. Part 3 -- uniqueness of weights,
varying the path, the degenerate affine Hecke algebras*. Notes from the lecture, covering Sections 3.2, 3.3, 4.1 in [RT1].

Feb 3, Lecture 4: *Representations of symmetric groups following Okounkov and Vershik. Part 4 (the last) -- irreducible
representations of the degenerate affine Hecke algebra H(2) and consequences. Completion of classification.*
Notes from the lecture, covering Sections 4.2, 4.3, 5.1 in [RT1].
Standard Young tableaux are covered only in [RT1], Section 5.2, please read the section.

We start a new topic: the representation theory of algebraic groups and Lie algebras. The list of references.

Feb 8, Lecture 5: *Representations of algebraic groups and Lie algebras, Part 1 -- a reminder on affine algebraic varieties.
Algebraic groups.* Notes including explanation of why rational representations are called so
(not covered in class) Updated on 2/12 to fix an inaccuracy in Sec 1.3. References: [OV], Section 3.1; [H2], Sections 7,8; [S], Section 2.1,2.2.

Feb 10, Lecture 6: * Representations of algebraic groups and Lie algebras, Part 2 -- tangent spaces in Algebraic geometry,
the bracket on the tanget space at 1 of an algebraic group*. Notes
including sketches of two equivalent constructions of the Lie algebra of an algebraic group. References: [H2], Sections 5.1, 9.1, [S],
Section 4.

Lecture 6.5: Bonus -- *Hopf algebras, distribution algebras for algebraic groups*. References: Basics of Hopf algebras are explained in many
books on quantum groups (e.g., Jantzen's book on quantum groups not to be confused in [J] or Kassel's book). Distribution algebras are discussed
e.g. in Section 7 of Part 1 of [J].

Feb 15, Lecture 7: *Representations of algebraic groups and Lie algebras, Part 3 --
Lie algebras and their representations, universal enveloping algebras and PBW theorem.* Notes. References: Section 2 in Chapter 1 and
Section 3 in Chapter 3 of [OV]; Chapter 3 in [K]; Sections 1 and 2 in [B]; Sections 1,2,17 in [H1].

Feb 17, Lecture 8: *Representations of algebraic groups and Lie algebras, Part 4 -- simple algebraic groups and Lie algebras.
Representation theory of sl_2 and SL_2 in characteristic 0, the classification of irreducibles. Bonus: algebraic subalgebras.*
Notes. References [H], Section 7; or [B], Sec 1 in Ch. 8; or [K], Sec 5.1.

Lecture 8.5: Bonus --*Distribution algebras, continued*.

Feb 22, Lecture 9: *Representations of algebraic groups and Lie algebras, Part 5 --
Representation theory of sl_2 and SL_2 in characteristic 0, the Casimir element and complete reducibility.
Representations of sl_2 in positive characteristic.* Notes.
Bonus: *p-center and p-central reductions*.
References: same as for Lecture 9 on characteristic 0 representations. References for characteristic p may appear
if I find something other than the original Rudakov-Shafarevich 1967 paper in Russian...

Feb 24, Lecture 10: *Representations of algebraic groups and Lie algebras, Part 6 --
representations of sl_2 and SL_2 in characteristic p.* Notes, includes
the complement section on the restricted pth power map. Mistake in Section 2 (some M(i)'s with i\geq p are irreducible!)
is fixed.
References may be provided later, content of Section 2 in a more general (and advanced) setting are in
Section 3 of Part II of [J].

Lecture 10.33: Bonus, *(x+y)^p-x^p-y^p is a Lie polynomial*.

Lecture 10.66: Bonus, *Frobenius kernel*.

Mar 1, Lecture 11: *Representations of algebraic groups and Lie algebras, Part 7 --
Representations of SL_2 in characteristic p, finished*. Notes (typo fixed 3/14).
References: Sec 2 of Part II in [J] does this in higher generality of connected reductive groups.

Mar 3, Lecture 12: *Representations of algebraic groups and Lie algebras, Part 8 --
Representations of sl_n in characteristic 0, classification of irreducibles*. Notes
with a complement carrying over the sl_n story to the other classical Lie algebras. References:
[K], Sections 9.1-9.3, [H1], Sections 20,21, [B], Section 7 in Chapter 8. Updated on 3/6 with some notation changed.

Mar 8, Lecture 13: *Representations of algebraic groups and Lie algebras, Part 9 -- Representations of sl_n in characteristic 0, the proof that L(\lambda) is finite dimensional
for dominant \lambda*. Notes. Bonus: Clifford algebras and spinor representations.
References: same as for Lec 12. Typos fixed 3/19.

Mar 10, Lecture 14: *The center of the universal enveloping algebra, the Harish-Chandra isomorphism, and complete reducibility*.
Notes.
References: [H1, Section 23], [B, Chapter 7, Section 8]. For different proofs of complete reducibility see
[B], Ch. 1, Sec. 6.5; [H1], Sec 6; [K], Sec 6.9. Typos fixed 3/19.

Lecture 14.5: Categorical representations of sl_2, part 1. Notes.

Mar 15, Lecture 15: *The Chevalley restriction theorem. Proof of the Harish-Chandra isomorphism. Characters.*
Notes, incl. complements on characters of rational representations and
on Schur polynomials. References. See the previous lecture for the HC isomorphism. [B], Ch. 7, Sec 9; [K], Sec. 9.5; [H1], Sec. 24
for the Weyl character formula. Typos fixed 3/19.

Mar 17, Lecture 16: *Proof of the Weyl character formula. What's next?*. Notes.
See the previous lecture for the Weyl character formula. Intro to Section 8 and Section 8.1 in [H3] for "What's next".

Mar 29, Lecture 17: *Rational representations of SL_n.* Notes. Proofs
for Facts in Section 1.2. Some references for the in-class discussion in Section 1.3.
References: [J], Sections 2 (for Section 1.1, 1.2 of the lecture) and 3-5 (for Section 1.3).

We start a new topic: Hecke algebra/category. The list of references (still to be completed).

Mar 31, Lecture 18: *Representations of GL_n(F_q) and the Hecke algebra*. Notes (with missing factor
fixed in 2) of Exercise in Section 2).

Apr 5, Lecture 19: *The generic Hecke algebra. Generalizations.* Notes. References include
[C],[L] for Section 1 and [Ka] for Section 2.

Apr 7, Lecture 20: *Kac-Moody algebras, cont'd. Weyl and Coxeter groups and their Hecke algebras.*
Notes (Section 2.4 wasn't covered in the lecture). References: Section 1 -- after the main theorem
and Section 4 in [Ka]; Section 2.1: Section 3 in [Ka]; Section 2.2: [B], various parts of Chapters 4-6; Section 2.3:
Chapters 1-3 in [Lu] (note a different normalization in Chapter 3 there).

Apr 12, Lecture 21: *Kazhdan-Lusztig basis in the Hecke algebra*. Notes. References:
[Lu], Chapters 4 and 5 for the general case of Hecke algebras with unequal parameters.

Apr 14, Lecture 22: *Bernstein-Gelfand-Gelfand category O* (a.k.a. the Hecke category starts to show up!).
Lecture 22 (with a long complement on variants and relatives of category O).
References: [Hu3] (ref from part 2), Sections 1.11, 1.12 for Sec 1, 3.8, for Sec 2.

Comments on Lectures 17-20.

Apr 19, Lecture 23: *Projectives in O, continued*. Notes. References: [H3], Sections 3.8, 7.

Apr 21, Lecture 24: *The category of projectives. The three theorems of Soergel*. Notes.
References: Section 1 -- [Hu3],3.7-3.11. Section 2 -- Mitya's (a.k.a. Dmytro's) lectures here.

Apr 26, Lecture 25: *Soergel (bi)modules*, notes. References: Mitya's and Borya's lectures for Sec 1.1, 1.4.
For Secs 1.2, 1.3, [EMTW], Secs 4.1,4.2,4.4-4.6.

Apr 28, Lecture 26 (a.k.a. the last but not really): *Soergel bimodules cont'd incl. the Soergel categorification theorem*.
Notes. References, [EMTW], Sec 4.

May 3, Lecture 27 (a.k.a. now finally the last, a.k.a. afterparty, a.k.a. one with a turtle):
* The geometry behind the Soergel theory: varieties and cohomology.*
Notes.

**Course afterlife:**
An account on the elephants (and the turtle)! An interpretation
(courtesy of Jack Miller and Pradeep).

Lecture 28A on why we should expect the Kazhdan-Lusztig conjecture to hold.

Lecture 29A, the 2nd part on actions of sl_2 (and more general Kac-Moody algebras) on categories.

Lecture 30A, (categorical) Hecke algebras and link invariants.

THE END!