Inna Entova: Deligne categories and stable Kronecker coefficients
Channel: Amritanshu Prasad
Date: 26/06/2020

In this talk, I will present an application of the theory of Deligne categories to the study of Kronecker coefficients.

Kronecker coefficients are structural constants for the category Rep(Sn)
of finite-dimensional representations of the symmetric group; namely, given three irreducible representations μ,τ,λ of Sn, the Kronecker coefficient Kron(λ,μ,τ) is the multiplicity of λ inside μ⊗τ. The study of Kronecker coefficients has been described as "one of the main problems in the combinatorial representation theory of the symmetric group", yet very little is known about them.

I will define a "stable" version of the Kronecker coefficients (due to Murnaghan), which generalizes both Kronecker coefficients and Littlewood-Richardson coefficients (structural constants for general linear groups).

It turns out that the stable Kronecker coefficients appear naturally as structural constants in the Deligne categories Rep(St), which are interpolations of the categories Rep(Sn) to complex t. I will explain this phenomenon and show that the categorical properties of Rep(St) allow us not only to recover known properties of the stable Kronecker coefficients, but also obtain new identities.

This is a report on my project from 2014.