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{\bf J. Novak}
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{\bf Truncations of Random Unitary Matrices and Young Tableaux}
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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.
\bye