Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials
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 Fq equipped 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 c...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України
2018
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| Zitieren: | Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials / J.-H. Lee, H. Tanaka // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. |
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Lee, J.-H. Tanaka, H. 2025-11-21T19:06:39Z 2018 Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials / J.-H. Lee, H. Tanaka // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05E30; 20C08; 33D45; 33D80 arXiv: 1709.07825 https://nasplib.isofts.kiev.ua/handle/123456789/209455 https://doi.org/10.3842/SIGMA.2018.009 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 Fq equipped 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 (C₁∨, 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. The authors thank Daniel Orr for helpful comments on nil-DAHAs of type (C∨n, 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]. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials |
| spellingShingle |
Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials Lee, J.-H. Tanaka, H. |
| title_short |
Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials |
| title_full |
Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials |
| title_fullStr |
Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials |
| title_full_unstemmed |
Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials |
| title_sort |
dual polar graphs, a nil-daha of rank one, and non-symmetric dual q-krawtchouk polynomials |
| author |
Lee, J.-H. Tanaka, H. |
| author_facet |
Lee, J.-H. Tanaka, H. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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 Fq equipped 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 (C₁∨, 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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209455 |
| citation_txt |
Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials / J.-H. Lee, H. Tanaka // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. |
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AT leejh dualpolargraphsanildahaofrankoneandnonsymmetricdualqkrawtchoukpolynomials AT tanakah dualpolargraphsanildahaofrankoneandnonsymmetricdualqkrawtchoukpolynomials |
| first_indexed |
2025-11-27T19:18:00Z |
| last_indexed |
2025-11-27T19:18:00Z |
| _version_ |
1850885966931165184 |