On the minor problem and branching coefficients

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. $\mathrm{SU}(n)$, into irreps of $U(n-1)$, resp. $\mathrm{SU}(n-1)$. As is well known, the branching coefficients are trivial (equal to $0$ or $1$) for the branchings of $U(n)\supset U(n-1)$, while they are not for $\mathrm{SU}(n)\supset \mathrm{SU}(n-1)$, where multiplicities may appear. In the latter case, the problem is shown to be related to the distribution of spacings in the minor problem. An explicit expression is obtained for the multiplicities, in terms of an integral stemming from the minor problem, and an Ansatz is given for a closed form expression for arbitrary $n$.