Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points
Abstract
In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type $A$ (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anticommuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters $\left( q,t\right) $ and are defined by means of a YangBaxter graph. The present paper determines the values of a subclass of the polynomials at the special points $\left( 1,t,t^{2},\ldots\right) $ or$\left( 1,t^{1},t^{2},\ldots\right) $. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve $\left( q,t\right)$hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.
 Publication:

Symmetry
 Pub Date:
 May 2021
 DOI:
 10.3390/sym13050779
 arXiv:
 arXiv:2104.00710
 Bibcode:
 2021Symm...13..779D
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Classical Analysis and ODEs;
 33D52;
 20C08;
 05E05
 EPrint:
 Symmetry 2021, 13 (5), 779