Journal of Combinatorics

Volume 3 (2012)

Number 3

Macdonald polynomials in superspace as eigenfunctions of commuting operators

Pages: 495 – 561

DOI: http://dx.doi.org/10.4310/JOC.2012.v3.n3.a8

Authors

Olivier Blondeau-Fournier (Département de physique, de génie physique et d’optique, Université Laval, Québec, Canada)

Patrick Desrosiers (Instituto de Matemática y Física, Universidad de Talca, Chile)

Luc Lapointe (Instituto de Matemática y Física, Universidad de Talca, Chile)

Pierre Mathieu (Département de physique, de génie physique et d’optique, Université Laval, Québec, Canada)

Abstract

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 a 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 Macdonald superpolynomials in terms of the Macdonald non-symmetric polynomials via a non-standard (anti)symmetrization and a suitable dressing by anticommuting monomials. This relationship naturally suggests the form of two families of commuting operators that have the defined superpolynomials as their common eigenfunctions. These eigenfunctions are then shown to be triangular and orthogonal. Up to a normalization, these two conditions uniquely characterize these superpolynomials. Moreover, the Macdonald superpolynomials are found to be orthogonal with respect to a second (constant-termtype) scalar product, and its norm is evaluated. The latter is shown to match (up to a $q$-power) the conjectured norm with respect to the original scalar product. Finally, we recall the super-version of the Macdonald positivity conjecture and present two new conjectures which both provide a remarkable relationship between the new $(q, t)$ Kostka coefficients and the usual ones.

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