41 research outputs found
A unified construction of generalised classical polynomials associated with operators of Calogero-Sutherland type
In this paper we consider a large class of many-variable polynomials which
contains generalisations of the classical Hermite, Laguerre, Jacobi and Bessel
polynomials as special cases, and which occur as the polynomial part in the
eigenfunctions of Calogero-Sutherland type operators and their deformations
recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present
a unified and explicit construction of all these polynomials
Observables of Macdonald processes
We present a framework for computing averages of various observables of
Macdonald processes. This leads to new contour--integral formulas for averages
of a large class of multilevel observables, as well as Fredholm determinants
for averages of two different single level observables.Comment: 36 pages, 1 figur
A unified construction of generalized classical polynomials associated with operators of calogero-Sutherland Type
In this paper we consider a large class of many-variable polynomials which contains generalizations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the eigenfunctions of Calogero–Sutherland type operators and their deformations recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present a unified and explicit construction of all these polynomials
Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory
We develop a new method for studying the asymptotics of symmetric polynomials
of representation-theoretic origin as the number of variables tends to
infinity. Several applications of our method are presented: We prove a number
of theorems concerning characters of infinite-dimensional unitary group and
their -deformations. We study the behavior of uniformly random lozenge
tilings of large polygonal domains and find the GUE-eigenvalues distribution in
the limit. We also investigate similar behavior for alternating sign matrices
(equivalently, six-vertex model with domain wall boundary conditions). Finally,
we compute the asymptotic expansion of certain observables in dense
loop model.Comment: Published at http://dx.doi.org/10.1214/14-AOP955 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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
Characters of classical groups, Schur-type functions, and discrete splines
We study a spectral problem related to the finite-dimensional characters of
the groups , , and , which form the classical series
, , and , respectively. The irreducible characters of these three
series are given by -variate symmetric polynomials. The spectral problem in
question consists in the decomposition of the characters after their
restriction to the subgroups of the same type but smaller rank . The main
result of the paper is the derivation of explicit determinantal formulas for
the coefficients in this decomposition.
In fact, we first compute these coefficients in a greater generality -- for
the multivariate symmetric Jacobi polynomials depending on two continuous
parameters. Next, we show that the formulas can be drastically simplified for
the three special cases of Jacobi polynomials corresponding to the --
characters. In particular, we show that then the coefficients are given by
piecewise polynomial functions. This is where a link with discrete splines
arises.
In type (that is, for the characters of the unitary groups ),
similar results were earlier obtained by Alexei Borodin and the author [Adv.
Math., 2012], and then reproved by another method by Leonid Petrov [Moscow
Math. J., 2014]. The case of the symplectic and orthogonal characters is more
intricate.Comment: Accepted in Sbornik: Mathematic
Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications
A unified approach is given to kernel functions which intertwine Ruijsenaars
difference operators of type A and of type BC. As an application of the
trigonometric cases, new explicit formulas for Koornwinder polynomials attached
to single columns and single rows are derived.Comment: 40 pages. Three sections are added in Appendix, as well as several
comments and reference