Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

TL;DR

This paper constructs subspaces of singular nonsymmetric Macdonald polynomials at special parameter values, demonstrating their irreducibility as Hecke algebra modules and ensuring pole-free coefficients at these points.

Contribution

It introduces a method to build pole-free subspaces of singular nonsymmetric Macdonald polynomials at specific parameters, revealing their irreducibility as Hecke algebra modules.

Findings

Subspaces of singular Macdonald polynomials are constructed at special $(q,t)$ values.

Coefficients of these polynomials have no poles at the specialized parameters.

The subspace forms an irreducible Hecke algebra module.

Abstract

Singular nonsymmetric Macdonald polynomials are constructed by use of the representation theory of the Hecke algebras of the symmetric groups. These polynomials are labeled by quasistaircase partitions and are associated to special parameter values $(q,t)$. For $N$ variables, there are singular polynomials for any pair of positive integers $m$ and $n$, with $2\leq n\leq N$, and parameters values $(q,t)$ satisfying $q^{a}t^{b}=1$ exactly when $a=rm$ and $b=rn$, for some integer $r$. The coefficients of nonsymmetric Macdonald polynomials with respect to the basis of monomials $\big\{ x^{\alpha}\big\}$ are rational functions of $q$ and $t$. In this paper, we present the construction of subspaces of singular nonsymmetric Macdonald polynomials specialized to particular values of $(q,t)$. The key part of this construction is to show the coefficients have no poles at the special values of $(q,t)$. Moreover, this subspace of singular Macdonald polynomials for the special values of the parameters is an irreducible module for the Hecke algebra of type $A_{N-1}$.