Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2020
• Main Authors: Colmenarejo, Laura, Dunkl, Charles F.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2020
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210600
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Singular Nonsymmetric Macdonald Polynomials and Quasistaircases. Laura Colmenarejo and Charles F. Dunkl. SIGMA 16 (2020), 010, 27 pages

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 with special parameter values (q,t). For N variables, there are singular polynomials for any pair of positive integers m and n, with 2 ≤ n ≤ N, and parameter values (q,t) satisfying qᵃtᵇ = 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 {xᵅ} 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 that 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 AN₋₁.