A hook-excess inequality for the support of Kronecker products of symmetric group characters

Abstract

We prove an inequality constraining which irreducible characters of the symmetric group S_n can occur in a tensor product. For a partition we count the cells outside its first hook, and the cells outside its first two rows and first two columns. If the Kronecker coefficient g(lambda, mu, nu) of three partitions of n is positive, then the number of cells of nu outside its double hook is at most the sum of the numbers of cells of lambda and mu outside their hooks. For two hooks this is the classical theorem of Remmel and Rosas, which the proof (an induction on n using the branching rule and Frobenius reciprocity) uses as base case. Consequences: a bound on the Durfee size of constituents, a necessary condition for a partition to have a universal tensor square, and the fact that no rectangle has one. Exhaustive computer checks for n up to 20 and a Lean 4 check of finite instances are reported. Status: preprint, not peer reviewed. The author used an AI assistant (Claude, Anthropic) for literature organisation, drafting, verification code and LaTeX preparation; the work has not been reviewed by an external specialist. Novelty was checked against the cited literature and zbMATH Open (October 2026) but not MathSciNet.Contact: sebastian.p.needer@gmail.com