© 1999 by London Mathematical Society
The Irreducible Specht Modules in Characteristic 2
Imperial College 180 Queen's Gate, London SW7 2BZ
University of Sydney Sydney, NSW 2006, Australia
Received 28 April 1998. Revision received 13 October 1998.
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 S
with a 2-regular partition were classified in [2]. Let ' denote the partition conjugate to . If S is irreducible, then S' is irreducible, since S' is isomorphic to the dual of S tensored with the sign representation. It turns out that if neither nor ' is 2-regular, then S is irreducible only if = (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.