Kronecker Products of sn-Characters in Hooks

Abstract

There are two products of symmetric group characters which have been studied extensively. One is the outer product, denoted ⊗. The outer product of an Sn-character and an Sm-character is an Sn+m-character. This product is relatively well-understood and can be calculated using the Littlewood– Richardson rule. The other character is the inner, or Kronecker product, denoted ⊗. The inner product can only be taken for two characters of the same symmetric group, Sn, and yields another Sn character. It is less wellunderstood and the algorithms that are used to compute the inner product are all more difficult to use than the Littlewood–Richardson rule. Two useful theorems on the inner product give upper bounds. Irreducible Sn-characters are naturally indexed by partitions of n. We write χ for the Sn-character indexed by the partition λ. Say λ, μ ∈ Par n and