Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 010, 27 pages      arXiv:1909.00071      https://doi.org/10.3842/SIGMA.2020.010

Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

Laura Colmenarejo ^a and Charles F. Dunkl ^b
a)  Department of Mathematics and Statistics, University of Massachusetts at Amherst, Amherst, USA
^b)  Department of Mathematics, University of Virginia, Charlottesville VA 22904-4137, USA

Received September 06, 2019, in final form February 19, 2020; Published online February 27, 2020

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 $A_{N-1}$.

Key words: nonsymmetric Macdonald polynomials; Dunkl operators; Hecke algebra; critical pairs.

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