Chapter 6 - Modular Representations

Summary

We turn now to the modular representation theory of symmetric groups; that is, we shall work over fields whose characteristic is a prime number p. There are two distinct ways of approaching the modular theory, either by working in terms of the characters or by way of a module-theoretic definition of the modular irreducibles. In this chapter we concentrate on the former, more classical approach. Here, the emphasis is on the p-block structure, and the central theorem, which is still called “Nakayama's conjecture” although it has long since been proved, provides a necessary and sufficient condition for two ordinary irreducible characters to lie in the same p-block. It is one of the most beautiful results in the entire theory of symmetric groups.

After presenting the theory of p-blocks, we shall look closely at the as yet unsolved problem of determining the decomposition matrices of S_n. It becomes clear in the end that some more powerful techniques are required, and we shall turn in the next chapter to one of the alternative ways of conducting an investigation of modular representations.

The p-block Structure of the Ordinary Irreducibles of S_n and A_n; Generalized Decomposition Numbers

Recall from 2.1.12 that every field is a splitting field for S_n and hence, in particular, ℤ_p, the field of p elements, is a splitting field. We may therefore restrict our attention to representations of S_n over ℤ_p, and call these the p-modular representations of S_n.