A bivariate Q-polynomial structure for the non-binary Johnson scheme

The notion of multivariate $P$- and $Q$-polynomial association scheme has been introduced recently, generalizing the well-known univariate case. Numerous examples of such association schemes have already been exhibited. In particular, it has been demonstrated that the non-binary Johnson scheme is a bivariate $P$-polynomial association scheme. We show here that it is also a bivariate $Q$-polynomial association scheme for some parameters. This provides, with the $P$-polynomial structure, the bispectral property (i.e. the recurrence and difference relations) of a family of bivariate orthogonal polynomials made out of univariate Krawtchouk and dual Hahn polynomials. The algebra based on the bispectral operators is also studied together with the subconstituent algebra of this association scheme.

[1]  L. Frappat,et al.  Representations of the rank two Racah algebra and orthogonal multivariate polynomials , 2022, Linear Algebra and its Applications.

[2]  J. Geronimo,et al.  Multivariable Askey–Wilson function and bispectrality , 2011 .

[3]  Hirotake Kurihara.  Character Tables of Association Schemes Based on Attenuated Spaces , 2011, 1101.3455.

[4]  Mizan Rahman,et al.  A Probablistic Origin for a New Class of Bivariate Polynomials , 2008, 0812.3879.

[5]  J. Geronimo,et al.  Bispectrality of Multivariable Racah–Wilson Polynomials , 2007, 0705.1469.

[6]  Paul-Hermann Zieschang,et al.  The exchange condition for association schemes , 2006 .

[7]  Rene F. Swarttouw,et al.  Orthogonal Polynomials , 2005, Series and Products in the Development of Mathematics.

[8]  R. A. Bailey.  Association Schemes: Designed Experiments, Algebra and Combinatorics , 2004 .

[9]  Paul M. Terwilliger,et al.  LEONARD PAIRS AND THE ASKEY-WILSON RELATIONS , 2003, math/0305356.

[10]  Paul M. Terwilliger.  Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an overview , 2001, math/0408390.

[11]  Paul M. Terwilliger,et al.  Some algebra related to P- and Q-polynomial association schemes , 2001, Codes and Association Schemes.

[12]  Paul M. Terwilliger.  The Subconstituent Algebra of an Association Scheme, (Part I) , 1992 .

[13]  A. Zhedanov.  “Hidden symmetry” of Askey-Wilson polynomials , 1991 .

[14]  M. V. Tratnik,et al.  Some multivariable orthogonal polynomials of the Askey tableau‐discrete families , 1991 .

[15]  B. Davies.  Onsager's algebra and superintegrability , 1990 .

[16]  Jean-Marie Goethals,et al.  On the Nonbinary Johnson Scheme , 1985, Eur. J. Comb..

[17]  D. A. Leonard.  Orthogonal Polynomials, Duality and Association Schemes , 1982 .

[18]  Michael Grady,et al.  Conserved charges from self-duality , 1982 .

[19]  B. C. Griffiths,et al.  Orthogonal Polynomials on the Multinomial Distribution , 1971 .

[20]  R. C. Bose,et al.  On Linear Associative Algebras Corresponding to Association Schemes of Partially Balanced Designs , 1959 .

[21]  L. Onsager.  Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[22]  P. Delsarte.  AN ALGEBRAIC APPROACH TO THE ASSOCIATION SCHEMES OF CODING THEORY , 2011 .

[23]  A. Hora,et al.  Distance-Regular Graphs , 2007 .

[24]  Paul M. Terwilliger,et al.  The Subconstituent Algebra of an Association Scheme (Part III) , 1993 .

[25]  Chris D. Godsil,et al.  ALGEBRAIC COMBINATORICS , 2013 .

[26]  Paul M. Terwilliger,et al.  The Subconstituent Algebra of an Association Scheme (Part II) , 1993 .