A Bijection for the Alperin Weight Conjecture in Sn

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 Sn, however there is no explicit bijection given between the two sets. In this paper we develop an explicit bijection between the p-weights of Sn and a certain set of partitions that is known to have the same cardinality as the irreducible Brauer characters of Sn. 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.