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Truncations of random unitary matrices and Young tableaux

Jonathan Novak
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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.

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Jonathan Novak. 2006-08-06. Truncations of random unitary matrices and Young tableaux. https://arxiv.org/abs/math/0608108

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