The target audience for this lecture is beginning Ph.D. students (I will give a related course at Yale in Fall 16).
Many properties of a finite group can be approached using formulas involving sums over characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group. In particular, the low dimensional representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might lead to proofs of some of these properties.
This talk will discuss a new method obtained with Roger Howe (Yale) for systematically constructing the small representations of finite classical groups. I will explain the method with concrete applications and a lot of numerical data that was generated together with John Cannon (Sydney) and Steve Goldstein (Madison).