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

SIGMA 14 (2018), 009, 27 pages      arXiv:1709.07825

Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual $q$-Krawtchouk Polynomials

Jae-Ho Lee a and Hajime Tanaka b
a) Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, USA
b) Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan

Received September 25, 2017, in final form January 29, 2018; Published online February 10, 2018

Let $\Gamma$ be a dual polar graph with diameter $D \geqslant 3$, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field $\mathbb{F}_q$ equipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt index $D$. From a pair of a vertex $x$ of $\Gamma$ and a maximal clique $C$ containing $x$, we construct a $2D$-dimensional irreducible module for a nil-DAHA of type $(C^{\vee}_1, C_1)$, 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 $\Gamma$ based on the type of the form.

Key words: dual polar graph; nil-DAHA; dual $q$-Krawtchouk polynomial; Terwilliger algebra; Leonard system.

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  1. Bannai E., Ito T., Algebraic combinatorics. I. Association schemes, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984.
  2. Brouwer A.E., Cohen A.M., Neumaier A., Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 18, Springer-Verlag, Berlin, 1989.
  3. Cherednik I., Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald's operators, Int. Math. Res. Not. 1992 (1992), 171-180.
  4. Cherednik I., Double affine Hecke algebras and Macdonald's conjectures, Ann. of Math. 141 (1995), 191-216.
  5. Cherednik I., Macdonald's evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), 119-145, q-alg/9503006.
  6. Cherednik I., Nonsymmetric Macdonald polynomials, Int. Math. Res. Not. 1995 (1995), 483-515, q-alg/9505029.
  7. Cherednik I., Orr D., One-dimensional nil-DAHA and Whittaker functions I, Transform. Groups 17 (2012), 953-987, arXiv:1104.3918.
  8. Cherednik I., Orr D., One-dimensional nil-DAHA and Whittaker functions II, Transform. Groups 18 (2013), 23-59, arXiv:1104.3918.
  9. Cherednik I., Orr D., Nonsymmetric difference Whittaker functions, Math. Z. 279 (2015), 879-938, arXiv:1302.4094.
  10. Van Dam E.R., Koolen J.H., Tanaka H., Distance-regular graphs, Electron. J. Combin. (2016), \#DS22, 156 pages, arXiv:1410.6294.
  11. Godsil C.D., Algebraic combinatorics, \textChapman and Hall Mathematics Series, Chapman & Hall, New York, 1993.
  12. Hosoya R., Suzuki H., Tight distance-regular graphs with respect to subsets, European J. Combin. 28 (2007), 61-74.
  13. Ito T., Tanabe K., Terwilliger P., Some algebra related to $P$- and $Q$-polynomial association schemes, in Codes and Association Schemes (Piscataway, NJ, 1999), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Vol. 56, Amer. Math. Soc., Providence, RI, 2001, 167-192, math.CO/0406556.
  14. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, \textSpringer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  15. Koornwinder T.H., Bouzeffour F., Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials, Appl. Anal. 90 (2011), 731-746, arXiv:1006.1140.
  16. Lee J.-H., $Q$-polynomial distance-regular graphs and a double affine Hecke algebra of rank one, Linear Algebra Appl. 439 (2013), 3184-3240, arXiv:1307.5297.
  17. Lee J.-H., Nonsymmetric Askey-Wilson polynomials and $Q$-polynomial distance-regular graphs, J. Combin. Theory Ser. A 147 (2017), 75-118, arXiv:1509.04433.
  18. Lee J.-H., Tanaka H., Dual polar graphs, a nil-DAHA of rank one, and non-symmetric dual $q$-Krawtchouk polynomials, Sém. Lothar. Combin. 78B (2017), Art. 42, 12 pages.
  19. Leonard D.A., Orthogonal polynomials, duality and association schemes, SIAM J. Math. Anal. 13 (1982), 656-663.
  20. Macdonald I.G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003.
  21. Mazzocco M., Non-symmetric basic hypergeometric polynomials and representation theory for confluent Cherednik algebras, SIGMA 10 (2014), 116, 10 pages, arXiv:1409.4287.
  22. Mazzocco M., Confluences of the Painlevé equations, Cherednik algebras and $q$-Askey scheme, Nonlinearity 29 (2016), 2565-2608, arXiv:1307.6140.
  23. Nomura K., Terwilliger P., The universal DAHA of type $(C_1^\vee,C_1)$ and Leonard pairs of $q$-Racah type, Linear Algebra Appl. 533 (2017), 14-83, arXiv:1701.06089.
  24. Oblomkov A., Stoica E., Finite dimensional representations of the double affine Hecke algebra of rank 1, J. Pure Appl. Algebra 213 (2009), 766-771, math.RT/0409256.
  25. Sahi S., Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. 150 (1999), 267-282, q-alg/9710032.
  26. Stanton D., Some $q$-Krawtchouk polynomials on Chevalley groups, Amer. J. Math. 102 (1980), 625-662.
  27. Stanton D., Three addition theorems for some $q$-Krawtchouk polynomials, Geom. Dedicata 10 (1981), 403-425.
  28. Suzuki H., The Terwilliger algebra associated with a set of vertices in a distance-regular graph, J. Algebraic Combin. 22 (2005), 5-38.
  29. Tanaka H., A bilinear form relating two Leonard systems, Linear Algebra Appl. 431 (2009), 1726-1739, arXiv:0807.0385.
  30. Tanaka H., Vertex subsets with minimal width and dual width in $Q$-polynomial distance-regular graphs, Electron. J. Combin. 18 (2011), no. 1, \#P167, 32 pages, arXiv:1011.2000.
  31. Terwilliger P., The subconstituent algebra of an association scheme. I, J. Algebraic Combin. 1 (1992), 363-388.
  32. Terwilliger P., The subconstituent algebra of an association scheme. II, J. Algebraic Combin. 2 (1993), 73-103.
  33. Terwilliger P., The subconstituent algebra of an association scheme. III, J. Algebraic Combin. 2 (1993), 177-210.
  34. Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001), 149-203, math.RA/0406555.
  35. Terwilliger P., Leonard pairs and the $q$-Racah polynomials, Linear Algebra Appl. 387 (2004), 235-276, math.QA/0306301.
  36. Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array, Des. Codes Cryptogr. 34 (2005), 307-332, math.RA/0306291.
  37. Worawannotai C., Dual polar graphs, the quantum algebra $U_q(\mathfrak{sl}_2)$, and Leonard systems of dual $q$-Krawtchouk type, Linear Algebra Appl. 438 (2013), 443-497, arXiv:1205.2144.

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