Stability of Kronecker coefficients via discrete tomography

TL;DR

This paper introduces a new additive stability condition for Kronecker coefficients, linking discrete tomography with representation theory, and unifies previous stability results under this framework.

Contribution

It establishes a novel connection between additivity of matrices and Kronecker coefficient stability, providing a simple way to generate new stability instances.

Findings

Additivity of matrices implies stability of Kronecker coefficients.

All previously known stability properties fit into the additive stability framework.

Additivity is easy to verify and produce many examples.

Abstract

In this paper we give a new sufficient condition for a general stability of Kronecker coefficients, which we call it additive stability. It was motivated by a recent talk of J. Stembridge at the conference in honor of Richard P. Stanley's 70th birthday, and it is based on work of the author on discrete tomography along the years. The main contribution of this paper is the discovery of the connection between additivity of integer matrices and stability of Kronecker coefficients. Additivity, in our context, is a concept from discrete tomography. Its advantage is that it is very easy to produce lots of examples of additive matrices and therefore of new instances of stability properties. We also show that Stembridge's hypothesis and additivity are closely related, and prove that all stability properties of Kronecker coefficients discovered before fit into additive stability.