  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.

Representation Theory;Algebraic Geometry