On the Minor Problem and Branching Coefficients

TL;DR

This paper revisits the Minor problem, exploring the spectrum of principal submatrices of Hermitian matrices under conjugation, and connects it with branching coefficients in representation theory.

Contribution

It introduces a new perspective on the Minor problem using orbital integrals and links it to the decomposition of irreducible representations of unitary groups.

Findings

Established connections between the spectrum of principal submatrices and branching coefficients.

Applied orbital integrals to analyze the Minor problem.

Enhanced understanding of representation decomposition in the context of Hermitian matrices.

Abstract

The Minor problem, namely the study of the spectrum of a principal submatrix of a Hermitian matrix taken at random on its orbit under conjugation, is revisited, with emphasis on the use of orbital integrals and on the connection with branching coefficients in the decomposition of an irreducible representation of U(n), resp. SU(n), into irreps of U(n-1), resp. SU(n-1).