On the Kronecker Product of Sn Characters

Abstract Let χ ρ , χ λ , χ μ be irreducible S n characters and assume χ ρ appears in the Kronecker product χ λ ⊗χ μ with maximal first part ρ 1 . Then ρ 1 = |λ∩μ| = ∑min(λ i , μ i ). A similar result holds for the maximal first column. We also give a recursive formula for χ λ ⊗χ μ . As an application, we show that if n = λ 1 + μ 1 − ρ 1 , then 〈χ λ ⊗χ μ χ ρ 〉 s n = 〈χ (λ2, λ3, ...) ⊗χ (μ2, μ3, ...) , χ (ρ2, ρ3, ...) 〉 s n − ρ 1 where ⊗ denotes the outer tensor product. These results are applied to study the character ∑χ λ ⊗χ λ 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.