Weights for symmetric and general linear groups

Abstract

An important feature of the theory of finite groups is the number of connections and analogies with the theory of Lie groups. The concept of a weight has long been useful in the modular representation theory of finite Lie groups in the defining characteristic of the group. The idea of a weight in the modular representation theory of an arbitrary finite group was recently introduced in Alperin ( Proc. Sympos. Pure Math. 41 (1987, 369–379), where it was conjectured that the number of weights should equal the number of modular irreducible representations. Moreover, this equality should hold block by block. The conjecture has created great interest, since its truth would have important consequences—a synthesis of known results and solutions of outstanding problems. In this paper we prove the conjecture first for the modular representations of symmetric groups and second for modular representations in odd characteristic r for the finite general linear groups. In the latter case r may be assumed to be different from the defining characteristic p of the group, since the result is known when r is p. The well-known analogy between the representation theory of the symmetric and general linear groups holds here too.