Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

TL;DR

This paper computes the values of certain nonsymmetric Macdonald superpolynomials at specific points, using algebraic induction and Hecke algebra techniques, revealing formulas involving hook products and special symmetry cases.

Contribution

It provides explicit evaluation formulas for a subclass of nonsymmetric Macdonald superpolynomials at special points, extending previous theoretical frameworks.

Findings

Explicit formulas for polynomial evaluations at special points.

Evaluation results involve (q,t)-hook products.

Includes evaluations for polynomials with restricted symmetry properties.

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 anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters $\left( q,t\right) $ and are defined by means of a Yang-Baxter 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.