19 - General and Special Linear Groups

Summary

In this chapter we look at some special features of representations of GL(n, q) and SL(n, q), most of which involve the natural action on the polynomial algebra S := K[x₁, …, x_n] (19.2–19.7). For example, the ring of invariants has been studied since the time of Dickson. In the case of SL(2, q) there is an interesting “periodicity” phenomenon (19.8), which shows a systematic pattern in the module structure of the homogeneous summands of S. The polynomial ring setting lends itself to combinatorial methods involving partitions, generating functions, and the like. Since algebraic group methods often remain in the background, we provide mainly a survey of the results with occasional proofs when these are self-contained.

To conclude the chapter, we discuss in 19.11 a method of passing from modular to ordinary representations called “Brauer lifting” which in a sense reverses the earlier procedure of reduction modulo p.

Although these topics can be studied in purely algebraic terms, they have many connections with problems in algebraic topology, involving especially the Steenrod algebra and classifying spaces: see for example Wilkerson and Henn–Schwartz. Often the prime p = 2 is most natural in these contexts, but the algebraic framework is more general.