The Irreducible Specht Modules in Characteristic 2

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 S 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 ) . 1991 Mathematics Subject Classification 20C30.