THE IRREDUCIBLE SPECHT MODULES IN CHARACTERISTIC 2

Abstract

In the representation theory of finite groups, it is useful to know which ordinary irreducible representations remain irreducible modulo a prime p. For the symmetric groups [Sfr ]_n, this amounts to determining which Specht modules are irreducible over a field of characteristic p. Throughout this note we work in characteristic 2, and in this case we classify the irreducible Specht modules, thereby verifying the conjecture in [3, p. 97].

Recall that a partition is 2-regular if all of its non-zero parts are distinct; otherwise the partition is 2-singular. The irreducible Specht modules S^λ with λ a 2-regular partition were classified in [2]. Let λ′ denote the partition conjugate to λ. If S^λ is irreducible, then S^λ′ is irreducible, since S^λ′ is isomorphic to the dual of S^λ tensored with the sign representation. It turns out that if neither λ nor λ′ is 2-regular, then S^λ is irreducible only if λ = (2, 2).

In order to state our theorem, we let l(k), for an integer k, be the least non-negative integer such that k<2^l(k).