Dual polar graphs, a nil-DAHA of rank one, and non-symmetric dual q-Krawtchouk polynomials
Document Type
Article
Publication Date
2-10-2018
Abstract
Let G be a dual polar graph with diameter D≥3, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field Fq equipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt index D. From a pair of a vertex x of T and a maximal clique C containing x, we construct a 2D-dimensional irreducible module for a nil-DAHA of type (Cv1,C1), and establish its connection to the generalized Terwilliger algebra with respect to x, C. Using this module, we then define the non-symmetric dual q-Krawtchouk polynomials and derive their recurrence and orthogonality relations from the combinatorial points of view. We note that our results do not depend essentially on the particular choice of the pair x, C, and that all the formulas are described in terms of q, D, and one other scalar which we assign to Г based on the type of the form.
Publication Title
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume
14
Digital Object Identifier (DOI)
10.3842/SIGMA.2018.009
E-ISSN
18150659
Citation Information
Lee, & Tanaka, H. (2018). Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials. Symmetry, Integrability and Geometry, Methods and Applications. https://doi.org/10.3842/SIGMA.2018.009