Macdonald polynomials in superspace as eigenfunctions of commuting operators

  title={Macdonald polynomials in superspace as eigenfunctions of commuting operators},
  author={Olivier Blondeau-Fournier and Patrick Desrosiers and Luc Lapointe and Pierre Mathieu},
  journal={arXiv: Mathematical Physics},
A generalization of the Macdonald polynomials depending upon both commuting and anticommuting variables has been introduced recently. The construction relies on certain orthogonality and triangularity relations. Although many superpolynomials were constructed as solutions of highly over-determined system, the existence issue was left open. This is resolved here: we demonstrate that the underlying construction has a (unique) solution. The proof uses, as a starting point, the definition of the… 

Tables from this paper

Nonsymmetric Macdonald Superpolynomials

  • C. Dunkl
  • Mathematics
    Symmetry, Integrability and Geometry: Methods and Applications
  • 2020
There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875]

Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

The values of a subclass of the polynomials at the special points 1,t,t2,… or 1, t−1,t−2,….

The N=2 supersymmetric Calogero-Sutherland model and its eigenfunctions

In a recent work, we have initiated the theory of N=2 symmetric superpolynomials. As far as the classical bases are concerned, this is a rather straightforward generalization of the N=1 case. However

Les polynômes de Macdonald dans le superespace et le modèle Ruijsenaars-Schneider supersymétrique

The theory of symmetric superpolynomials ([DLM03, DLM06]) is further extended with the introduction of a family of superpolynomials that depends upon two parameters, denoted by q and t. This new

N≥ symmetric superpolynomials

The theory of symmetric functions has been extended to the case where each variable is paired with an anticommuting one. The resulting expressions, dubbed superpolynomials, provide the natural N=1

Multi-Macdonald polynomials

On the Hopf algebra of noncommutative symmetric functions in superspace

We study in detail the Hopf algebra of noncommutative symmetric functions in superspace sNSym , introduced by Fishel, Lapointe and Pinto. We introduce a family of primitive elements of sNSym and

Schur Superpolynomials: Combinatorial Definition and Pieri Rule

Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the

$m$-Symmetric functions, non-symmetric Macdonald polynomials and positivity conjectures

We study the space, Rm, of m-symmetric functions consisting of polynomials that are symmetric in the variables xm+1, xm+2, xm+3, . . . but have no special symmetry in the variables x1, . . . , xm. We



Macdonald Polynomials in Superspace: Conjectural Definition and Positivity Conjectures

We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested

Jack Polynomials in Superspace

AbstractThis work initiates the study of orthogonal symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the


he Macdonald polynomials with prescribed symmetry are obtained from the non-symmetric Macdonald polynomials via the operations of t-symmetrization, t-antisymmetrization and normalization. Motivated

Orthogonality of Jack polynomials in superspace

Classical symmetric functions in superspace

Various basic results, such as the generating functions for the multiplicative bases, Cauchy formulas, involution operations as well as the combinatorial scalar product are also generalized.

Algebraic Methods and 𝑞-Special Functions

Science fiction and Macdonald's polynomials by F. Bergeron and A. M. Garsia On the expansion of elliptic functions and applications by R. Chouikha Generalized hypergeometric functions-Classification

A new scalar product for nonsymmetric Jack polynomials

Symmetric Jack polynomials arise naturally in several contexts, including statistics, physics, combinatorics, and representation theory. They are pairwise orthogonal with repsect to two different

A Normalization Formula for the Jack Polynomials in Superspace and an Identity on Partitions

We prove a conjecture of Desrosiers, Lapointe and Mathieu giving a closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly

A q-analogue of the type A Dunkl operator and integral kernel

We introduce the $q$-analogue of the type $A$ Dunkl operators, which are a set of degree--lowering operators on the space of polynomials in $n$ variables. This allows the construction of