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 Sn, 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 Sλ with λ a 2-regular partition were classified in [2]. Let λ′ denote the partition conjugate to λ. If Sλ is irreducible, then Sλ′ is irreducible, since Sλ′ is isomorphic to the dual of Sλ tensored with the sign representation. It turns out that if neither λ nor λ′ is 2-regular, then Sλ 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 < 2l(k). 1991 Mathematics Subject Classification 20C30.