On the asymptotics of Kronecker coefficients, 2

TL;DR

This paper explores the geometric structure of stable triples of Kronecker coefficients, revealing that they form unions of faces of the moment polytope with varying dimensions, advancing understanding of their asymptotic behavior.

Contribution

It extends previous geometric methods to characterize the set of stable triples as unions of faces of the moment polytope, including those of codimension one.

Findings

Stable triples form unions of faces of the moment polytope.

Faces of stable triples can have different dimensions.

Many faces of stable triples have codimension one.

Abstract

Kronecker coefficients encode the tensor products of complex irreducible representations of symmetric groups. Their stability properties have been considered recently by several authors (Vallejo, Pak and Panova, Stembridge). In previous works we described a geometric method, based on Schur-Weyl duality, that allows to produce huge series of instances of this phenomenon. In this note we show how to go beyond these so-called additive triples. We show that the set of stable triples defines a union of faces of the moment polytope. Moreover these faces may have different dimensions, and many of them have codimension one.