On the Kronecker Product of Sn Characters Original Research Article

Let χρ, χλ, χμ be irreducible Sn characters and assume χρ appears in the Kronecker product χλcircle times operatorχμ 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 χλcircle times operatorχμ. As an application, we show that if n = λ1 + μ1 − ρ1, then left angle bracketχλcircle times operatorχμχρright-pointing angle bracketsn = left angle bracketχ(λ2, λ3, ...)circle times operatorχ(μ2, μ3, ...), χ(ρ2, ρ3, ...) right-pointing angle bracketsn − ρ1 where circle times operator denotes the outer tensor product. These results are applied to study the character ∑χλcircle times operatorχλ 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.