The Irreducible Specht Modules in Characteristic 2

In the representation theory of nite 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 eld 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 with a 2{regular partition were classiied in 2]. Let 0 denote the partition conjugate to. If S is irreducible then S 0 is irreducible, since S 0 is isomorphic to the dual of S tensored with the sign representation. It turns out that if neither nor 0 is 2{regular then S is irreducible only if = (2; 2). In order to state our theorem, for an integer k, we let l(k) be the least non{negative integer such that k < 2 l(k) .