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 → ∞. 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.are the Catalan numbers. This estimate has been known to physicists for some time, see e.g.[1] for an application to quantum transport in mesoscopic systems.The first proof of (1.2) is due to Collins [4], and is quite involved. The appearance of the Catalan numbers, which are ubiquitous in enumerative combinatorics [44, Exercise 6.19], motivates the search for combinatorial structure underlying the averages u 11 . . . u nn , u 1π(1) . . . u nπ(n) N which might lead to a conceptually simple proof of (1.2). Along these lines, it was observed in [33] that the evaluation of the basic inner products is intimately related to a certain "inverse problem" in the character theory of the symmetric groups. This connection is explored in the present article. Working on the symmetric group side, we obtain a number of results concerning this "inverse problem" which give a new perspective on the structure of polynomial integrals on the unitary group (leading in particular to a new proof of (1.2)) and are moreover of significant interest in their own right.

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

“…In this subsection, we review some results in [19]. These should be compared with theorems given in the next subsection.…”

“…. , J n ) is an element of the center of the group algebra in C[S n ], see, e.g., [8,19]. We define the coefficients L λ μ (n) for the monomial symmetric function m λ via…”

“…Matsumoto and Novak [19] gave a combinatorial explicit expression of A μ (m λ , n) with |λ| = |μ|, where m λ is the monomial symmetric function.…”

“…. , N} and N ≥ n. The Weingarten calculus for unitary groups developed in [3,5,23] states that those integrals are given by a sum of Weingarten functions In [20] (see also [19]), a remarkable connection between Wg (−1) …”

Combinatorial Aspects of Orthogonal Group Integrals

“…More results here were obtained by Collins and Matsumoto in [10], and Zinn-Justin [22]. These combinatorial methods have a number of concrete applications, notably to random matrix questions [6], [9], [11], [12], [16], [18], [20], and to invariance questions [5].…”