Truncations of a Random Unitary Matrix and Young Tableaux

Abstract. Let U be a matrix chosen randomly, with respect to Haar measure, from the unitary group U(d). We express the moments of the trace of any submatrix of U as a sum over partitions whose terms count certain standard and semistandard Young tableaux. Using this combinatorial interpretation, we obtain a simple closed form for the moments of an individual entry of a random unitary matrix and use this to deduce that the entries converge in moments to standard complex Gaussian random variables. In addition, we recover a well-known theorem of E. Rains which shows that the moments of the trace of a random unitary matrix enumerate permutations with restricted increasing subsequence length.