The Heisenberg product seen as a branching problem for connected reductive groups, stability properties

TL;DR

This paper explores the Heisenberg product in complex symmetric group representations, revealing its connection to branching coefficients in reductive groups and establishing stability properties using geometric invariant theory.

Contribution

It demonstrates that Aguiar coefficients are branching coefficients for connected reductive groups, enabling geometric methods to analyze their stability.

Findings

Aguiar coefficients are branching coefficients for reductive groups

Established stability properties of Aguiar coefficients

Extended previous results using geometric invariant theory

Abstract

In this article we study, in the context of complex representations of symmetric groups, some aspects of the Heisenberg product, introduced by Marcelo Aguiar, Walter Ferrer Santos, and Walter Moreira in 2017. When applied to irreducible representations, this product gives rise to the Aguiar coefficients. We prove that these coefficients are in fact also branching coefficients for representations of connected complex reductive groups. This allows to use geometric methods already developped in a previous article, notably based on notions from Geometric Invariant Theory, and to obtain some stability results on Aguiar coefficients, generalising some of the results concerning them given by Li Ying.