Dimensions of components of tensor products of the linear groups representations with applications to Beurling-Fourier algebras

TL;DR

This paper establishes universal bounds on the sizes of components in tensor products of linear group representations, with applications to harmonic analysis and Beurling-Fourier algebras.

Contribution

It provides new universal bounds on the dimensions of tensor product components for GL(n) and SL(n) representations, advancing harmonic analysis applications.

Findings

Universal bounds on isotypic component dimensions for GL(n)

Universal bounds on irreducible component dimensions for SL(n)

Applications to Beurling-Fourier algebra theory

Abstract

We give universal upper bounds on the relative dimensions of isotypic components of a tensor product of the linear group GL(n) representations and universal upper bounds on the relative dimensions of irreducible components of a tensor product of the special linear group SL(n) representations. This problem is motivated by harmonic analysis problems, and we give some applications of this result in the theory of Beurling-Fourier algebras.