Truncations of Random Unitary Matrices and Young Tableaux

Abstract

Let $U$ be a matrix chosen randomly, with respect to Haar measure, from the unitary group $U(d).$ For any $k \leq d,$ and any $k \times k$ submatrix $U_k$ of $U,$ we express the average value of $|{\rm Tr}(U_k)|^{2n}$ as a sum over partitions of $n$ with at most $k$ rows whose terms count certain standard and semistandard Young tableaux. We combine our formula with a variant of the Colour-Flavour Transformation of lattice gauge theory to give a combinatorial expansion of an interesting family of unitary matrix integrals. In addition, we give a simple combinatorial derivation of the moments of a single entry of a random unitary matrix, and hence deduce that the rescaled entries converge in moments to standard complex Gaussians. Our main tool is the Weingarten function for the unitary group.