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

Abstract

We prove that if the Kronecker coefficient g(λ, μ, ν) of three partitions of n is positive, then the number of cells of ν outside its first two rows and first two columns is at most the sum of the numbers of cells of λ and of μ outside their first hooks. For two hooks this is the classical theorem of Remmel and Rosas, which our induction uses as its base case; we also give a computer-assisted proof of that base case (an exact rational certificate checked by an independent program) and a Lean 4 formalisation for the Specht modules of every symmetric group, whose only non-standard ingredient is that one certificate check is done by Lean's compiled evaluation (native_decide). The same induction, with a containment theorem of Berele and Regev as base case, gives a family of inequalities indexed by pairs of positive integers; this part is a short written argument that has not been formalised and has not been reviewed. As consequences we bound the Durfee size of the constituents and obtain a necessary condition for a partition to have a universal tensor square. We report exhaustive computations for n ≤ 21 and compare the inequality with Dvir's bounds, the depth bound and the vanishing criterion of Young and Vallejo: it certifies about 2-4% of the classes of zeros for n ≤ 15, but only about 0.2-0.3% beyond the Young-Vallejo criterion, almost all of these lying inside the Kronecker cone, so that no homogeneous linear inequality valid on the support can detect them. For n = 23 we exhibit twelve classes of triples with g = 0, computed exactly, that satisfy the hypothesis of the inequality and are certified by none of the criteria we tested; we have no general description of such triples. The paper was written with substantial AI assistance (Claude, Anthropic). The proofs have not undergone formal peer review, and we do not claim novelty beyond the searches we list. This is version 3; the same version is archived on Zenodo (DOI 10.5281/zenodo.23549266).

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