## 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.