Jack symmetric functions and some combinatorial properties of young symmetrizers

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27452/1/0000492.pd

A Markov chain on the symmetric group and Jack symmetric functions

Diaconis and Shahshahani studied a Markov chain W[function of (italic small f)](1) whose states are the elements of the symmetric group S[function of (italic small f)]. In W[function of (italic small f)](1), you move from a permutation [pi] to any permutation of the form [pi](i, j) with equal probability. In this paper we study a deformation W[function of (italic small f)]([alpha]) of this Markov chain which is obtained by applying the Metropolis algorithm to W[function of (italic small f)](1). The stable distribution of W[function of (italic small f)]([alpha]) is [alpha][function of (italic small f)]-c([pi]) where c([pi]) denotes the number of cycles of [pi]. Our main result is that the eigenvectors of the transition matrix of W[function of (italic small f)]([alpha]) are the Jack symmetric functions. We use facts about the Jack symmetric functions due to Macdonald and Stanley to obtain precise estimates for the rate of convergence of W[function of (italic small f)]([alpha]) to its stable distribution.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30109/1/0000481.pd

Moments of the eigenvalue densities and of the secular coefficients of β-ensembles

This is an Open Access Article. It is published by IOP under the Creative Commons Attribution 4.0 Unported Licence (CC BY). Full details of this licence are available at: http://creativecommons.org/licenses/by/4.0/© 2017 IOP Publishing Ltd & London Mathematical Society.We compute explicit formulae for the moments of the densities of the eigenvalues of the classical β-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In particular, the moments are linear combinations of averages of Jack polynomials, whose coefficients are related to specific examples of Jack characters

A note on the map expansion of Jack polynomials

In a recent work, Maciej Do\l{}e\k{}ga and the author have given a formula of the expansion of the Jack polynomial $J^{(\alpha)}_\lambda$ in the power-sum basis as a non-orientability generating series of bipartite maps whose edges are decorated with the boxes of the partition $\lambda$. We conjecture here a variant of this expansion in which we restrict the sum on maps whose edges are injectively decorated by the boxes of $\lambda$. We prove this conjecture for Jack polynomials indexed by 2-column partitions. The proof uses a mix of combinatorial methods and differential operator computations.Comment: 20 pages, 2 figure

Moments of the eigenvalue densities and of the secular coefficients of β-ensembles

© 2017 IOP Publishing Ltd & London Mathematical Society.We compute explicit formulae for the moments of the densities of the eigenvalues of the classical β-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In particular, the moments are linear combinations of averages of Jack polynomials, whose coefficients are related to specific examples of Jack characters

Zonal polynomials via Stanley's coordinates and free cumulants

We study zonal characters which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. We show that the zonal characters, just like the characters of the symmetric groups, admit a nice combinatorial description in terms of Stanley's multirectangular coordinates of Young diagrams. We also study the analogue of Kerov polynomials, namely we express the zonal characters as polynomials in free cumulants and we give an explicit combinatorial interpretation of their coefficients. In this way, we prove two recent conjectures of Lassalle for Jack polynomials in the special case of zonal polynomials.Comment: 45 pages, second version, important change