Name Herr Dr. Ajay Ramadoss Adresse c/o Andrea WaldburgerRämistrasse 101, HG G44.1Forschungsinstitut für Mathematik 8092 ZürichSWITZERLAND Departement Mathematik Beziehung Akademischer Gast

NummerTitelECTSUmfangDozierende
401-3160-12LRepresentation Theory of Associative Algebras6 KP2SG. Felder, A. Ramadoss
KurzbeschreibungIntroduction 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.
LernzielThe 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.
Inhalt1) 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.
Extensions.
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
[D]

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
[BGP]
Literatur[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