Related papers: Non-Symmetric Basic Hypergeometric Polynomials and…
In this paper, we study the polynomial representation of the double affine Hecke algebra of type $(C^\vee_n, C_n)$ for specialized parameters. Inductively and combinatorially, we give a linear basis of the representation in terms of linear…
We introduce and study deformations of finite-dimensional modules over rational Cherednik algebras. Our main tool is a generalization of usual harmonic polynomials for Coxeter groups -- the so-called quasiharmonic polynomials. A surprising…
We investigate the representation theory of the rational and trigonometric Cherednik algebra of type $GL_n$ by means of combinatorics on periodic (or cylindrical) skew diagrams. We introduce and study standard tableaux and plane partitions…
We introduce an explicit family of representations of the double affine Hecke algebra $\mathbb{H}$ acting on spaces of quasi-polynomials, defined in terms of truncated Demazure-Lusztig type operators. We show that these quasi-polynomial…
The quadratic algebras Q_n are associated with pseudo-roots of noncommutative polynomials. We compute the Hilbert series of the algebras Q_n and of the dual quadratic algebras Q_n^!
We review properties of the $q-$Hermite polynomials and indicate their links with the Chebyshev, Rogers--Szeg\"{o}, Al-Salam--Chihara, continuous $q-$% utraspherical polynomials. In particular we recall the connection coefficients between…
The q-Hermite I-Sobolev type polynomials of higher order are consider for their study. Their hypergeometric representation is provided together with further useful properties such as several structure relations which give rise to a…
We provide an algebraic interpretation for two classes of continuous $q$-polynomials. Rogers' continuous $q$-Hermite polynomials and continuous $q$-ultraspherical polynomials are shown to realize, respectively, bases for representation…
Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero-Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the…
In this article, we exhaustively explore the terminating basic hypergeometric representations and transformations of the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials. These subfamilies are obtained by repeatedly…
Infinitesimal Cherednik algebras, first introduced in [EGG], are continuous analogues of rational Cherednik algebras, and in the case of gl_n, are deformations of universal enveloping algebras of the Lie algebras sl_{n+1}. Despite these…
Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations $L_c(\lambda^\pm)$ of the rational Cherednik algebra…
We study the polynomial representation of the rational Cherednik algebra of type $A_{n-1}$ with generic parameter in characteristic $p$ for $p \mid n$. We give explicit formulas for generators for the maximal proper graded submodule, show…
We give a non-negative combinatorial formula, in terms of Littlewood-Richardson numbers, for the homology of the unitary representations of the cyclotomic rational Cherednik algebra, and as a consequence, for the graded Betti numbers for…
We give an overview of the representation theory of restricted rational Cherednik algebras. These are certain finite-dimensional quotients of rational Cherednik algebras at t=0. Their representation theory is connected to the geometry of…
This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category O. For type A, we explain relations with the…
We develop representation theory of the rational Cherednik algebra H associated to a finite Coxeter group W in a vector space h. It is applied to show that, for integral values of parameter `c', the algebra H is simple and Morita equivalent…
Classification of finite dimensional representations of the q-deformed Heisenberg algebra $H_q(3)$ is made by the help of Clifford algebra of polynomials and generalized Grassmann algebra. Special attention is paid when $q$ is a primitive…
In this note we explicitly construct an action of the rational Cherednik algebra $H_{1,m/n}(S_n,\mathbb{C}^n)$ corresponding to the permutation representation of $S_n$ on the $\mathbb{C}^{*}$-equivariant homology of parabolic Hilbert…
We prove a generalization of the Hermitian version of the Helton-Vinnikov determinantal representation of hyperbolic polynomials to the class of semi-hyperbolic polynomials, a strictly larger class, as shown by an example. We also prove…