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arXiv · 0905.1992

Jucys-Murphy Elements and Unitary Matrix Integrals

Sho Matsumoto·Jonathan Novak
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Abstract

In this paper, we study the relationship between polynomial integrals on the unitary group and the conjugacy class expansion of symmetric functions in Jucys-Murphy elements. Our main result is an explicit formula for the top coefficients in the class expansion of monomial symmetric functions in Jucys-Murphy elements, from which we recover the first order asymptotics of polynomial integrals over $\U(N)$ as $N \rightarrow \infty$. Our results on class expansion include an analogue of Macdonald's result for the top connection coefficients of the class algebra, a generalization of Stanley and Olshanski's result on the polynomiality of content statistics on Plancherel-random partitions, and an exact formula for the multiplicity of the class of full cycles in the expansion of a complete symmetric function in Jucys-Murphy elements. The latter leads to a new combinatorial interpretation of the Carlitz-Riordan central factorial numbers.

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Sho Matsumoto, Jonathan Novak. 2012-02-07. Jucys-Murphy Elements and Unitary Matrix Integrals. https://arxiv.org/abs/0905.1992

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