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Suppose you have a finite group G and you want to study certain related structures (e.g., random walks, Cayley graphs, word maps, etc.). In many cases, this might be done using Fourier type sums over the characters of G. A serious obstacle in applying these sums is lack of knowledge on the low dimensional representations of G. In fact, numerics shows that 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.

The “standard" method to construct representations of finite classical group is due to Deligne and Lusztig (1976). However, it seems that their approach has relatively little to say about the “small" representations.

This talk, aimed for beginning graduate students, discuss a joint project (see arXiv:1609.01276) with Roger Howe (Yale). We introduce a language to speak about “size” of a representation, and we develop a method for systematically construct (conjecturally all the) “small" representations of finite classical groups. The method is closely related to the mathematics of quantum mechanics.

I will illustrate our theory with concrete motivations and numerical data obtained with John Cannon (MAGMA, Sydney) and Steve Goldstein (Scientific computing, Madison).