Suppose you have a finite group G and you want to study certain related structures (random walks, Cayley graphs, word maps, etc.). In some cases, this might be done using sums over the characters of G. A serious obstacle in applying these formulas seemed to be lack of knowledge over the low dimensional representations of G. In fact, the “small representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might lead to proofs of some important conjectures. Despite the classification by Lusztig of the irreducible representations of finite groups of Lie type, it seems that this aspect remains obscure.
This talk will discuss a joint project with Roger Howe (Yale), where we introduce a language to define, and a new method for systematically construct, the small representations of finite classical groups.
I will demonstrate our theory with concrete motivations and numerical data obtained with John Cannon (MAGMA, Sydney) and Steve Goldstein (Scientific computing, Madison).
Talk sponsored by the Mrs. Hepsa Ely Silliman Memorial Fund.