Jucys-Murphy Elements and Unitary Matrix Integrals

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 $N\to \infty$. 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.

30 pages, final version, to appear in IMRN

References in corpus (4)

• On some properties of orthogonal Weingarten functions
• Unitary Integrals and Related Matrix Models
• Rate of convergence of linear functions on the unitary group
• Diagonal approximation of the form factor of the unitary group

Cited by in corpus (7)

• Jucys-Murphy elements, orthogonal matrix integrals, and Jack measures
• On polynomial integrals over the orthogonal group
• The orthogonal Weingarten formula in compact form
• Primitive factorizations, Jucys-Murphy elements, and matrix models
• On complete functions in Jucys-Murphy elements
• Integration on complex Grassmannians, deformed monotone Hurwitz numbers, and interlacing phenomena
• Procesi's Conjecture on the Formanek-Weingarten Function is False