6 - The q-Schur algebra

Summary

The representation theory of the finite general linear groups was investigated throughout the nineteen-eighties, the cases of the describing and non-describing characteristic being attacked with equal vigour. Fong and Srinivasan [1982] produced an elegant classification of the p-block structure of GL_n (q) where (p, q) = 1. Their labours produced a rule similar to the Nakayama Rule, 5.1.1, thus demonstrating that the case of positive non-describing characteristic could be as interesting as the case of the natural characteristic. James [1984] unearthed plentiful evidence for the assertion that the modular representation theory of Σ_n is ‘the case q = 1’ of this coprime theory.

Following this line, Dipper and James (independently and also in collaboration) unleashed the q-Schur algebra, S_q(n, r), on an unsuspecting world. The q-Schur algebras are related to the Hecke algebras associated to G = Σ_r in much the same way as are Schur algebras and group algebras of symmetric groups. Moreover, embeds into if we take q to be a prime power and this enables us to handle the representations of the finite reductive group. It is quite astonishing that both the representations theory of symmetric groups and the representation theory of the infinite general linear groups over K, discussed in previous chapters, are special cases of the representation theory of KGL_n(q), where n any integer such that n ≥ 2 and where q is a prime power coprime to the characteristic of K (so that q is a root of unity mod p).