401-3160-12L  Representation Theory of Associative Algebras

Semester Spring Semester 2012
Lecturers G. Felder, A. Ramadoss
Periodicity non-recurring course
Language of instruction English

Abstract Introduction to representation theory with many examples. Lie algebras and universal enveloping algebra. Schur lemma, representations of a matrix algebra. Jordan-Holder theorem, extensions. Category O for sl(2). Representations of finite groups. Burnside theorem, Frobenius reciprocity. Representations of symmetric groups. Representations of GL_2(F_q). Quivers. McKay correspondence.
Objective The presentations will introduce the main notions and results and illustrate them by working out the basic examples. We will also attempt to solve as many exercise problems from the book as possible.
The students are expected to read the book during the semester and give a talk on one of the subjects. There will be an online wiki associated to the seminar. Each student will be responsible for writing the solution of a few exercises and is expected to participate in the discussion to improve solutions to other exercises.
Content 1) Basic notions of representation theory of algebras.
Irreducible and indecomposable representations.
Schur lemma.
Irreducible representations of commutative algebras.
Representations of C[x] and Jordan normal form.
Cyclic representations.
[E] 2.2-2.6

2) Examples of algebras and their representations.
Weyl algebra, path algebras, Lie algebras, their representation, universal enveloping algebra.
Duals and tensor products.
Representations of sl(2).
[E] 2.8 - 2.9, 2.14 - 1.16

3) General results of representation theory I.
Representations of a matrix algebra.
Density theorem.
Semisimple algebras.
The group algebra of a finite group is a semisimple algebra.
[E] 3.1 - 3.6

4) General results of representation theory II.
Jordan-Holder theorem.
Krull-Schmidt theorem.
Representation of tensor products.
[E] 3.7-3.10, 8.1-8.2

5) Representations of finite groups: basic results.
Mashke's theorem, regular representation. characters.
Representations of quaternions, Dihedral groups, S_3,S_4,A_4.
[E] 4.1 -- 4.10,4.12
[F] 1.3,2.3

6) Representations of finite groups: further results
Frobenius-Schur indicator.
Algebraic integers, Burnside theorem.
Frobenius divisibility.
[E] 5.1-- 5.5

7) Induction and Restriction.
Frobenius reciprocity.
[E] 5.6-5.11

8) Representations of symmetric groups.
Combinatorics of representations of S_n.
Young diagrams, Young tableaux, Specht modules,
hook-length formula
[E] 5.12-5.17
[F] 4

9) Representation of general linear groups.
Schur-Weyl duality, algebraic representations of GL(V), representations of GL_2(F_q)
[E] 5.18-5.25
[F] 5.2

10) Quiver representation I
Dynkin diagrams,
MacKay graphs and
finite sugroups of SU(2)
[E] 6.1

11) Quiver representations II
low dimensions:
representations of A_1,A_2,A_3 and D_4
roots, Weyl group, Coxeter groups
[E] 6.2 -- 6.4
[S] V.1-V.6

12) Gabriel's theorem
Roots, Reflection functors, Gabriel's theorem.
[E] 6.5 -- 6.9
Literature [E] P.Etingof et al, Introduction to representation theory, AMS, available at the Polybuchandlung http://www.polybuchhandlung.ch
Much of the material (but not the historical interludes) can be found at http://math.mit.edu/~etingof/replect.pdf

Additional literature that might be helpful:
[F] W.Fulton, Representation theory, A first course.
[L] S. Lang, Algebra.
[S] J-P. Serre, Lie Algebras and Lie Groups
[BGP] I N Bernstein, I M Gel'fand and V A Ponomarev "COXETER FUNCTORS AND GABRIEL'S THEOREM"
Russ. Math. Surv. 28
[D] Igor Dolgachev
McKay's correspondence for cocompact discrete subgroups of SU(1,1)
available at http://arxiv.org/pdf/0710.2253
Prerequisites / Notice Prerequisites: linear algebra and basic notions of algebra. Please refresh (or learn) basic notions of multilinear algebra to be able to solve the first problems on tensor products of vector spaces in [E].