Nonsymmetric Macdonald Superpolynomials

TL;DR

This paper extends the theory of nonsymmetric Macdonald polynomials to superpolynomials valued in anti-commuting variables, providing new bases, orthogonality properties, and explicit norm formulas within the Hecke algebra framework.

Contribution

It introduces and studies nonsymmetric Macdonald superpolynomials in anti-commuting variables, combining representation theory and orthogonality properties for the first time.

Findings

Construction of orthogonal bases for superpolynomials

Explicit squared norm formulas for Macdonald superpolynomials

Factorization formulas at special evaluation points

Abstract

There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [S\'em. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here "symmetric" means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, and some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials there is a factorization when the polynomials are evaluated at special points.