Horn inequalities for nonzero Kronecker coefficients

Abstract

The Kronecker coefficients gαβγ and the Littlewood- Richardson coefficients cαβγ are nonnegative integers depending on three partitions α, β, and γ. By definition, gαβγ (resp. cαβγ) are the multiplicities of the tensor product decomposition of two irreducible representations of symmetric groups (resp. linear groups). By a classical Littlewood-Murnaghan's result the Kronecker coefficients extend the Littlewood-Richardson ones.The nonvanishing of the Littlewood-Richardson coefficient cαβγ implies that (α,β,γ) satisfies some linear inequalities called Horn inequalities. In this paper, we extend the essential Horn inequalities to the triples of partitions corresponding to a nonzero Kronecker coefficient.Along the way, we describe the set of tripless (α,β,γ) of partitions such that cαβγ≠0 and l(α)≤e, l(β)≤f and l(γ)≤e+f, for some given positive integers e and f. This set is the natural analogue of the classical Horn semigroup when one thinks about cαβγ as the branching multiplicities for the subgroup GLe×GLf of GLe+f.