A Geometric Approach to the stabilisation of certain sequences of Kronecker coefficients

TL;DR

This paper presents a geometric proof for the stabilization of certain Kronecker coefficient sequences, providing explicit bounds and extending the approach to other representation-theoretic coefficients.

Contribution

It offers a new geometric invariant theory-based proof and explicit bounds for Kronecker coefficient stabilization, also applicable to plethysm and hyperoctahedral group coefficients.

Findings

Explicit geometric bounds for stabilization

Application to Murnaghan's stability

Extension to plethysm and hyperoctahedral coefficients

Abstract

We give another proof, using tools from Geometric Invariant Theory, of a result due to S. Sam and A. Snowden in 2014, concerning the stability of Kro-necker coefficients. This result states that some sequences of Kronecker coefficients eventually stabilise, and our method gives a nice geometric bound from which the stabilisation occurs. We perform the explicit computation of such a bound on two examples, one being the classical case of Murnaghan's stability. Moreover, we see that our techniques apply to other coefficients arising in Representation Theory: namely to some plethysm coefficients and in the case of the tensor product of representations of the hyperoctahedral group.