On the asymptotics of Kronecker coefficients

TL;DR

This paper introduces a geometric approach based on Schur-Weyl duality to analyze the stability and asymptotic behavior of Kronecker coefficients, providing explicit computations and geometric insights.

Contribution

It presents a new geometric method to produce and analyze stable Kronecker coefficients, including explicit polytope descriptions and connections to affine Dynkin diagrams.

Findings

Generated large series of stable Kronecker coefficients

Computed stable coefficients as points in explicit polytopes

Connected rectangular Kronecker coefficients to affine Dynkin diagram E6

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). We describe a geometric method, based on Schur-Weyl duality, that allows to produce huge series of instances of this phenomenon. Moreover the method gives access to lots of extra information. Most notably, we can often compute the stable Kronecker coefficients, sometimes as number of points in very explicit polytopes. We can also describe explicitely the moment polytope in the neighbourhood of our stable triples. Finally, we explain an observation of Stembridge on the behaviour of certain rectangular Kronecker coefficients, by relating it to the affine Dynkin diagram of type $E\_6$.