Some reducible Specht modules

Abstract

We wish to consider which ordinary irreducible representations of the symmetric group Sn remain irreducible modulo a prime p; this is the same as asking which partitions λ of n have the property that the corresponding Specht module S is reducible over a field of characteristic p. If λ or its conjugate partition λ′ is p-regular then the answer is known [6,8]; it is also known in the case p = 2 [7]. This paper discusses some of the reducible Specht modules in the case that p 3. Throughout,p will be an odd prime and λ a partition of some integer n. We will prove some cases of a conjecture by James and Mathas [9], given below. We begin with some definitions.