On the asymptotics of Kronecker coefficients

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 numbers of points in very explicit polytopes. We can also describe explicitly 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 $$E_6$$E6.