Stability of Kronecker coefficients via discrete tomography

Abstract

For partitions λ, μ, ν of size n, the Kronecker coefficient g(λ,μ,ν) is the multiplicity of the irreducible complex character χν of the symmetric group Sn in the Kronecker product χλ⊗χμ. About eighty years ago Murnaghan found the first stability property of Kronecker coefficients. Recently Stembridge introduced the notion of stable triple in order to study different instances of stability of Kronecker coefficients and stated two conjectures. In this paper we use the notion of additivity, that first appeared in discrete tomography, to disprove one of them. We also show that additivity implies Stembridge’s condition for a triple of partitions to be stable. In this way we produce several new examples of stable triples. As a byproduct of the interplay of ideas between representation theory and discrete tomography, we obtain a new characterization of additivity.