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.
We provide a compact proof of the recent formula of Collins and Matsumoto for the Weingarten matrix of the orthogonal group using Jucys-Murphy elements.
We present a method to compute the class expansion of a symmetric function in the Jucys-Murphy elements of the symmetric group. We apply this method to one-row Hall-Littlewood symmetric functions, which interpolate between power sums and complete symmetric functions.
Consider the elements of the group algebra CS_{n} given by R_{j}=Sigma_{i=1}^{j-1}(ij), for 2<=j<=n. Jucys [3 - 5] and Murphy[7] showed that these elements act diagonally on elements of S_{n} and gave explicit formulas for the diagonal entries. As requested by the late S. Kerov, we give a combinatorial proof of this work in case j=n and present several similar results which arise from these combinatorial methods.
A factorization of a permutation into transpositions is called "primitive" if its factors are weakly ordered. We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting connections with Jucys-Murphy elements, symmetric group characters, and matrix models are described.
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.
A connection is made between complete homogeneous symmetric polynomials in Jucys-Murphy elements and the unitary Weingarten function from random matrix theory. In particular we show that $h_r(J_1,...,J_n),$ the complete homogeneous symmetric polynomial of degree $r$ in the JM elements, coincides with the $r$th term in the asymptotic expansion of the Weingarten function. We use this connection to determine precisely which conjugacy classes occur in the class basis resolution of $h_r(J_1,...,J_n),$ and to explicitly determine the coefficients of the classes of minimal height when $r < n.$ These …
In this paper, we look at the number of factorizations of a given permutation into star transpositions. In particular, we give a natural explanation of a hidden symmetry, answering a question of I.P. Goulden and D.M. Jackson. We also have a new proof of their explicit formula. Another result is the normalized class expansion of some central elements of the symmetric group algebra introduced by P. Biane. To obtain this results, we use natural analogs of Jucys-Murphy elements in the algebra of partial permutations of V. Ivanov and S. Kerov. We investigate their properties and use a formula of A.…
The problem of computing the class expansion of some symmetric functions evaluated in Jucys-Murphy elements appears in different contexts, for instance in the computation of matrix integrals. Recently, M. Lassalle gave a unified algebraic method to obtain some induction relations on the coefficients in this kind of expansion. In this paper, we give a simple purely combinatorial proof of his result. Besides, using the same type of argument, we obtain new simpler formulas. We also prove an analogous formula in the Hecke algebra of $(S_{2n},H_n)$ and use it to solve a conjecture of S. Matsumoto o…
Although powers of the Young-Jucys-Murphya elements X_i = (1 i) + ... +(i-1 i), i = 1, ..., n, in the symmetric group S_n acting on {1, ...,n} do not lie in the centre of the group algebra of S_n, we show that transitive powers, namely the sum of the contributions from elements that act transitively on {1, >...,n}, are central. We determine the coefficients, which we call star factorization numbers, that occur in the resolution of transitive powers with respect to the class basis of the centre of S_n, and show that they have a polynomiality property. These centrality and polynomiality properti…
G. E. Murphy showed in 1983 that the centre of every symmetric group algebra has an integral basis consisting of a specific set of monomial symmetric polynomials in the Jucys--Murphy elements. While we have shown in earlier work that the centre of the group algebra of S_3 has exactly three additional such bases, we show in this paper that the centre of the group algebra of S_4 has infinitely many bases consisting of monomial symmetric polynomials in Jucys--Murphy elements, which we characterize completely. The proof of this result involves establishing closed forms for coefficients of class su…
We describe a recursive algorithm that produces an integral basis for the centre of the Hecke algebra of type A consisting of linear combinations of monomial symmetric polynomials of Jucys--Murphy elements. We also discuss the existence of integral bases for the centre of the Hecke algebra that consist solely of monomial symmetric polynomials of Jucys--Murphy elements. Finally, for n=3, we show that only one such basis exists for the centre of the Hecke algebra, by proving that there are exactly four bases for the centre of the corresponding symmetric group algebra which consist solely of mono…
We give a new version of the fusion procedure for the symmetric group which originated in the work of Jucys and was developed by Cherednik. We derive it from the Jucys-Murphy formulas for the diagonal matrix units for the symmetric group.
We exhibit a vertex operator which implements multiplication by power-sums of Jucys-Murphy elements in the centers of the group algebras of all symmetric groups simultaneously. The coefficients of this operator generate a representation of ${\cal W}_{1+\infty}$, to which operators multiplying by normalized conjugacy classes are also shown to belong. A new derivation of such operators based on matrix integrals is proposed, and our vertex operator is used to give an alternative approach to the polynomial functions on Young diagrams introduced by Kerov and Olshanski.
Many aspects of the asymptotics of Plancherel distributed partitions have been studied in the past fifty years, in particular the limit shape, the distribution of the longest rows, connections with random matrix theory and characters of the representation matrices of the symmetric group. Regarding the latter, we expand a celebrated result of Kerov on the asymptotic of Plancherel distributed characters by studying partial trace and partial sum of a representation matrix. We decompose these objects into a main term and a reminder, proving a central limit theorem for both main terms and a law of …
Parametric families in the centre ${\bf Z}({\bf C}[S_n])$ of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda $τ$-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sph…
We study the asymptotic behavior of the free cumulants (in the sense of free probability theory of Voiculescu) of Jucys--Murphy elements--or equivalently--of the transition measure associated with a Young diagram. We express these cumulants in terms of normalized characters of the appropriate representation of the symmetric group S_q. Our analysis considers the case when the Young diagrams rescaled by q^{-1/2} converge towards some prescribed shape. We find explicitly the second order asymptotic expansion and outline the algorithm which allows to find the asymptotic expansion of any order. As …
The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily: to each conjugacy class we associate a two-dimensional surface and the asymptotic properties of the conjugacy class depend only on the genus of the resulting surface. This construction closely resembles …
This is an overview of recent results on the use of 2D Toda $τ$-functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an isomorphism between the zero charge sector of the Fermionic Fock space and the direct sum of the centers of the group algebra of the symmetric groups $S_n$. Specializing the fermionic formula to the case of diagonal group elements gives $τ$-functions of hypergeometric type, for which the expansion over products of Schur functions is diagonal, with coefficients of {\…
The central object of study is a formal power series that we call the content series, a symmetric function involving an arbitrary underlying formal power series $f$ in the contents of the cells in a partition. In previous work we have shown that the content series satisfies the KP equations. The main result of this paper is a new partial differential equation for which the content series is the unique solution, subject to a simple initial condition. This equation is expressed in terms of families of operators that we call $\mathcal{U}$ and $\mathcal{D}$ operators, whose action on the Schur sym…
The set $X$ of $k$-subsets of an $n$-set has a natural graph structure where two $k$-subsets are connected if and only if the size of their intersection is $k-1$. This is known as the Johnson graph. The symmetric group $S_n$ acts on the space of complex functions on $X$ and this space has a multiplicity-free decomposition as sum of irreducible representations of $S_n$, so it has a well-defined Gelfand-Tsetlin basis up to scalars. The Fourier transform on the Johnson graph is defined as the change of basis matrix from the delta function basis to the Gelfand-Tsetlin basis. The direct application…
