Jucys-Murphy elements, orthogonal matrix integrals, and Jack measures

Abstract

We study symmetric polynomials whose variables are odd-numbered Jucys-Murphy elements. They define elements of the Hecke algebra associated to the Gelfand pair of the symmetric group with the hyperoctahedral group. We evaluate their expansions in zonal spherical functions and in double coset sums. These evaluations are related to integrals of polynomial functions over orthogonal groups. Furthermore, we give an extension of them, based on Jack polynomials.

36 pages. v2: minor correction

References in corpus (4)

• On some properties of orthogonal Weingarten functions
• Jack polynomials and free cumulants
• Jucys-Murphy elements and Weingarten matrices
• Jucys-Murphy Elements and Unitary Matrix Integrals

Cited by in corpus (15)

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• Weingarten calculus for matrix ensembles associated with compact symmetric spaces
• Integration of invariant matrices and application to statistics
• General moments of matrix elements from circular orthogonal ensembles
• On Kerov polynomials for Jack characters
• Asymptotics for products of characteristic polynomials in classical $\beta$-Ensembles
• $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
• Primitive factorizations, Jucys-Murphy elements, and matrix models
• Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations
• Moments of a single entry of circular orthogonal ensembles and Weingarten calculus
• Generators of the Hecke algebra of $(S_{2n},B_n)$
• Commutators of random matrices from the unitary and orthogonal groups
• Random stochastic matrices from classical compact Lie groups and symmetric spaces
• On complete functions in Jucys-Murphy elements
• Polynomiality of shifted Plancherel averages and content evaluations