5 research outputs found
Logarithmic and complex constant term identities
In recent work on the representation theory of vertex algebras related to the
Virasoro minimal models M(2,p), Adamovic and Milas discovered logarithmic
analogues of (special cases of) the famous Dyson and Morris constant term
identities. In this paper we show how the identities of Adamovic and Milas
arise naturally by differentiating as-yet-conjectural complex analogues of the
constant term identities of Dyson and Morris. We also discuss the existence of
complex and logarithmic constant term identities for arbitrary root systems,
and in particular prove complex and logarithmic constant term identities for
the root system G_2.Comment: 26 page
Jack vertex operators and realization of Jack functions
We give an iterative method to realize general Jack functions from Jack
functions of rectangular shapes. We first show some cases of Stanley's
conjecture on positivity of the Littlewood-Richardson coefficients, and then
use this method to give a new realization of Jack functions. We also show in
general that vectors of products of Jack vertex operators form a basis of
symmetric functions. In particular this gives a new proof of linear
independence for the rectangular and marked rectangular Jack vertex operators.
Thirdly a generalized Frobenius formula for Jack functions was given and was
used to give new evaluation of Dyson integrals and even powers of Vandermonde
determinant.Comment: Expanded versio
On absolute moments of characteristic polynomials of a certain class of complex random matrices
Integer moments of the spectral determinant of complex
random matrices are obtained in terms of the characteristic polynomial of
the Hermitian matrix for the class of matrices where is a
given matrix and is random unitary. This work is motivated by studies of
complex eigenvalues of random matrices and potential applications of the
obtained results in this context are discussed.Comment: 41 page, typos correcte
Asymptotics for products of characteristic polynomials in classical -Ensembles
We study the local properties of eigenvalues for the Hermite (Gaussian),
Laguerre (Chiral) and Jacobi -ensembles of random matrices.
More specifically, we calculate scaling limits of the expectation value of
products of characteristic polynomials as . In the bulk of the
spectrum of each -ensemble, the same scaling limit is found to be
whose exact expansion in terms of Jack polynomials is well
known. The scaling limit at the soft edge of the spectrum for the Hermite and
Laguerre -ensembles is shown to be a multivariate Airy function, which
is defined as a generalized Kontsevich integral. As corollaries, when
is even, scaling limits of the -point correlation functions for the three
ensembles are obtained. The asymptotics of the multivariate Airy function for
large and small arguments is also given. All the asymptotic results rely on a
generalization of Watson's lemma and the steepest descent method for integrals
of Selberg type.Comment: [v3] 35 pages; this is a revised and enlarged version of the article
with new references, simplified demonstations, and improved presentation. To
be published in Constructive Approximation 37 (2013
q-Selberg Integrals and Macdonald Polynomials
Using the theory of Macdonald polynomials, a number of q-integrals of Selberg type are proved