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 - Funding Information:
The authors thank Daniel Orr for helpful comments on nil-DAHAs of type (Cn: Cn). They also thank Paul Terwilliger for many valuable discussions, Marta Mazzocco and Alexei Zhedanov for bringing the authors’ attention to [21, 22], and the anonymous referees for carefully reading the paper. Part of this work was done while Jae-Ho Lee was visiting Tohoku University as a JSPS Postdoctoral Fellow. Hajime Tanaka was supported by JSPS KAKENHI Grant Numbers JP25400034 and JP17K05156. An extended abstract of this work appeared in the proceedings of FPSAC’17, London, UK, July 2017; cf. [18].
Funding Information:
The authors thank Daniel Orr for helpful comments on nil-DAHAs of type (Cn∨, Cn). They also thank Paul Terwilliger for many valuable discussions, Marta Mazzocco and Alexei Zhedanov for bringing the authors’ attention to [21, 22], and the anonymous referees for carefully reading the paper. Part of this work was done while Jae-Ho Lee was visiting Tohoku University as a JSPS Postdoctoral Fellow. Hajime Tanaka was supported by JSPS KAKENHI Grant Numbers JP25400034 and JP17K05156. An extended abstract of this work appeared in the proceedings of FPSAC’17, London, UK, July 2017; cf. [18].
Publisher Copyright:
© 2018, Institute of Mathematics. All rights reserved.

PY - 2018/2/10

Y1 - 2018/2/10

N2 - 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.

AB - 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.

KW - Dual polar graph

KW - Dual q-krawtchouk polynomial

KW - Leonard system

KW - Nil-DAHA

KW - Terwilliger algebra

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U2 - 10.3842/SIGMA.2018.009

DO - 10.3842/SIGMA.2018.009

M3 - Article

AN - SCOPUS:85042052531

VL - 14

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

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

SN - 1815-0659

M1 - 009

ER -