On the Kronecker Product of Sn Characters

AbstractLet χρ, χλ, χμ be irreducible Sn characters and assume χρ appears in the Kronecker product χλ⊗χμ with maximal first part ρ1. Then ρ1 = |λ∩μ| = ∑min(λi, μi). A similar result holds for the maximal first column. We also give a recursive formula for χλ⊗χμ. As an application, we show that if n = λ1 + μ1 − ρ1, then 〈χλ⊗χμχρ〉sn = 〈χ(λ2, λ3, ...)⊗χ(μ2, μ3, ...), χ(ρ2, ρ3, ...) 〉sn − ρ1 where ⊗ denotes the outer tensor product. These results are applied to study the character ∑χλ⊗χλ where λ runs through the partitions with no more then k parts. This character is closely related to the polynomial identities of the algebra of k × k matrices