MATH 7313, Representation theory: modern introduction (Fall 2015)
Unofficial syllabus. An official one can be found on
Blackboard.
Sept 9, Lecture 0, notes: I recall a bunch of basic facts about the representation theory
of associative algebras and finite groups that often appear in graduate algebra classes.
Sept 14, Lecture 1 notes: I start to explain an approach of Okounkov and Vershik to the representation
theory of symmetric groups. The approach is inductive in nature: what one needs to understand is
how irreducible representations of S_n restrict to S_m\subset S_n.
Sept 16, Lecture 2 notes: I continue (and hopefully finish) the discussion of the representation theory of the symmetric
groups. In particular, we are going to see how degenerate affine Hecke algebras naturally appear in this theory.
Problem Set 1, due Sept 30,
Hints, Solutions.
Sept 21, Lecture 3 notes: I start to discuss algebraic groups and their Lie algebras. Then I study the representation
theory of sl_2(C). I assume that the audience knows basic definitions related to Lie algebras.
Class moves: 4.30-6pm, 544NI
Sept 23, Lecture 4 notes: I discuss the representations of sl_2(F) and SL_2(F), where F is an algebraically closed
field of positive characteristic.
Sept 28, Lecture 5 notes, picture:
I explain the structure theory of semisimple Lie algebras in characteristic 0.
Sept 30, Lecture 6 notes: I introduce Kac-Moody algebras. Then I discuss the finite dimensional representation theory
of semisimple Lie algebras in characteristic 0. And, finally, I discuss the structure and representation
theory of reductive algebraic groups in characteristic 0.
Problem Set 2, due Oct 19,
Hints, Solutions.
Oct 5, Lecture 7 notes: we discuss the category O for a semisimple Lie algebra in characteristic 0. Then we switch to representations
of reductive algebraic groups, both in 0 and positive characteristic.
Oct 7, Lecture 8 notes: we discuss the representation theory of semisimple Lie algebras in positive characteristic. Then we start a new topic:
we study the complex representations of GL_n(F_q).
Oct 14, Lecture 9 notes (include Deligne-Lusztig induction, not to be covered
in class): we discuss the Hecke algebras for general Weyl groups and the structure and representation
theory of general finite groups of Lie type, hopefully, including the Deligne-Lusztig induction.
Oct 19, Lecture 10 notes: I define Kazhdan-Lusztig bases in Hecke algebras and explain its connection to multiplicities
in categories O.
Oct 21, Lecture 11 notes: I continue to study the category O and introduce Soergel (bi)modules.
Problem Set 3, due Nov 4,
Hints, Solutions.
Oct 26, Lecture 12 notes: I talk about Hopf algebras and define the quantum group U_q(sl_2).
Oct 28, Lecture 13 notes: I will discuss the representation theory of U_q(sl_2) when q is not a root of 1 and also
the universal R-matrix.
Nov 2, Lecture 14 notes, picture 1,
picture 2, picture 3
: I explain how to get link invariants from representations of quantum groups.
Nov 4, Lecture 15 notes (with corrected formulas!): I discuss the quantum groups at roots of 1.
Problem Set 4, due Nov 25,
Hints, Solutions.
Class on Monday, Nov 9 is cancelled, and Nov 11 is a holiday
Nov 16, Lecture 16 notes, picture: I will discuss the Kac theorem on the indecomposable representations of quivers.
I will also start deformed preprojective algebras.
Nov 18, Lecture 17 notes: I will discuss the representations of deformed preprojective algebras and their application to Kac's theorem.
Nov 23, Lecture 18 notes: I will talk about an application of the deformed preprojective algebras to
the additive Deligne-Simpson problem.
Nov 24, 4.30-5.40, 509 Lake, Lecture 18.5,
hand-written notes: I will talk about a connection between deformed preprojective
algebras and Kleinian singularities.
Problem Set 5, due Dec 14, 12pm,
Hints, Solutions.
Nov 30, Lecture 19 notes: I start discussing actions of Kac-Moody algebras on categories and the representation theory of the symmetric
groups in positive characteristic.
Dec 2, Lecture 20 notes: I continue from last time giving a formal definitin of a categorical action and discribing K_0
of the modules over the symmetric group in positive characteristic.
Dec 7, Lecture 21 hand-written notes: We discuss categorical actions on the category O for gl_n and its parabolic versions.
This is the last lecture!!! On Wednesday, Dec 9, there will be lectures by Joel Kamnitzer.