TY - JOUR

T1 - Dual polar graphs, a nil-DAHA of rank one, and non-symmetric dual q-krawtchouk polynomials

AU - Lee, Jae Ho

AU - Tanaka, Hajime

N1 - Publisher Copyright:
Copyright © 2017, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2017/9/22

Y1 - 2017/9/22

N2 - Let Γ 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 Fqequipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt index D. From a pair of a vertex x of Γ 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 Γ based on the type of the form.MSC Codes 05E30, 20C08, 33D45, 33D80

AB - Let Γ 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 Fqequipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt index D. From a pair of a vertex x of Γ 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 Γ based on the type of the form.MSC Codes 05E30, 20C08, 33D45, 33D80

KW - Dual polar graph

KW - Dual q-Krawtchouk polynomial

KW - Leonard system

KW - nil-DAHA

KW - Terwilliger algebra

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