Basic Preliminary Concepts for A. Ram's Jan 26 Talks

Nantel Bergeron

York University

The basic preliminary concepts for Arun Ram's talk are:

1) Irreducible representations are indexed by shapes

2) Skew shape representations also make sense

3) Bases of the irreducible representations and skew shape representations are indexed by standard Young tableaux

4) The Littlewood-Richardson rule tells how a skew shape representation decomposes

5) Finite Coxeter groups are generalizations of symmetric groups and are controlled by hyperplane arrangements

6) Nobody knows much about the combinatorics of irreducible representations of finite Coxeter groups (except in types A,B,D, which are essentially all done by doing type A and then fiddling with it to get the others)

The proofs of these things are not important, just the ideas and definitions.

It might help if people had seen the presentations of finite Coxeter groups in terms of simple reflections.

Also, we should point out that for us irreducible representation means simple module. We hardly ever work with matrices when we talk about this stuff, always vector spaces with actions.