#### Blocks and Counting Conjectures

##### Presenter

February 4, 2008

**Keywords:**

- modular representation theory
- finite group theory
- irreducible vs. indecomposable representations
- p-regular orbits
- Jordan-Holder theorem
- finite representation type
- socle and (Jacobson) radical
- Wedderburn theory
- projective modules
- Cartan matrix (variation)
- block idempotents
- defect groups
- Donovan's conjecture
- Morita equivalence
- Brauer's theorem on blocks
- Fong-Reynolds theorem
- Brauer pair

**MSC:**

- 20-xx
- 20C05
- 20Cxx
- 20C20
- 20Dxx
- 20D10
- 20D25
- 20D20
- 20D30
- 16K50

##### Abstract

The workshop will focus on surveying main active areas of representation theory of finite groups, especially highlighting major unsolved problems. It is meant to be accessible for graduate students and non-specialists with some background in representation theory. The bulk of the week's program will be four short series of lectures: Course 1. Block theory and counting conjectures. The course will introduce the basic ideas of modular representations, including block theory, the main theorems of Brauer and the Green correspondence. Special theories for cyclic and nilpotent blocks will be covered. Subsequently, several counting conjectures will be discussed. These include the Alperin-McKay conjecture, Alperin's weight conjecture, the Knorr-Robinson synthesis via alternating sums, Dade's conjecture and recent subtle refinements. Course 2. Representation theory of groups of Lie type. While emphasizing the general linear group, this course covers topics including representations in characteristic zero, p and "el" and related structures such as Hecke algebras. Course 3. Representation theory and topology. The purpose of this course is to describe some of the important tie-ins between representation theory and algebraic topology through topics from cohomology of groups applied to representation theory, homological algebra (e.g. derived categories), fusion systems and p-local finite groups. Course 4. Broue's abelian defect group conjecture. This course will focus on equivalences between derived categories of blocks and on Broue's isotypies between blocks. In the case of finite groups of Lie type, related geometric structures enter the picture. These include Deligne-Lusztig varieties and complex reflection groups. As another illuminating example, the case of the symmetric groups will be discussed. The four courses will be supplemented by a number of single lectures on a variety of topics.