The stability of the Kronecker products of Schur functions

TL;DR

This paper investigates the stabilization phenomenon of Kronecker products of Schur functions, determining exact stabilization points and improving bounds for the stabilization of Kronecker coefficients.

Contribution

It precisely computes the stabilization point for Kronecker coefficients and introduces improved bounds, advancing understanding of their asymptotic behavior.

Findings

Exact stabilization value of n for Kronecker products

Two new bounds for coefficient stabilization

Improved bounds over previous results by Brion and Vallejo

Abstract

In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n for which all the coefficients of a Kronecker product of Schur functions stabilize. We also compute two new bounds for the stabilization of a sequence of coefficients and show that they improve existing bounds of M. Brion and E. Vallejo.