A Bijection for the Alperin Weight Conjecture in S n

Abstract

The Alperin weight conjecture states that if G is a finite group and p is a prime, then the number of irreducible Brauer characters of a group G should be equal to the number of conjugacy classes of p-weights of G. This conjecture is known to be true for the symmetric group S n , however there is no explicit bijection given between the two sets. In this paper we develop an explicit bijection between the p-weights of S n and a certain set of partitions that is known to have the same cardinality as the irreducible Brauer characters of S n . We also develop some properties of this bijection, especially in relation to a certain class of partitions whose corresponding Specht modules over fields of characteristic p are known to be irreducible.