Abstract:
This talk is based on the two joint papers [1] and [2] with Hirotake  Kurihara (Yamaguchi University), Da Zhao (East China University of Science  and Technology) and Yan Zhu (Shanghai University for Science and Technology).

The classification problem of P-and Q-polynomial association schemes  has been one of the very important problems in algebraic combinatorics.  The theorem of Leonard (1982) that the spherical functions as well as  their character tables are expressed by Askey-Wilson polynomials and their  special cases or relatives (i.e., limiting cases) was the important starting point.

These polynomials are one variable orthogonal polynomials. It seems that there  have been many studies of multivariable generalizations of Askey-Wilson polynomials  (at the level of orthogonal polynomials), but it seems that multivariable version of  P-and Q-polynomial association schemes (i.e., higher rank P-and Q-polynomial  association schemes) has not been studied much except for some very special cases.

Recently, the first very successful attempt was made by Bernard-N Crampé-d’Andecy-Vinet-Zaimi [3] for bivariate P-polynomial (or Q-polynomial) association schemes.  Then, motivated by [3], we obtained in [1] the concept of multivariate P-polynomial  (or Q-polynomial) association schemes for any monomial order.

The purpose of this talk is first to review the development of the  study of P-and/or Q-polynomial association schemes of higher rank following [1],[3].  Then we will discuss some explicit examples of multivariate P-and Q-polynomial  association schemes following [1], [2], [3], [4], [5] and so on.  Finally, we will  give some speculations concerning to which direction this study should proceed.

[1] E. Bannai, H. Kurihara, D. Zhao, Y. Zhu, Multivariate
P-and/or Q-polynomial association schemes, arXiv:2305.00707v2.

[2] E. Bannai, H. Kurihara, D. Zhao, Y. Zhu, Bivariate Q-polynomial structures for the
nonbinary Johnson scheme and the association scheme obtained from attenuated spaces, arXiv:2403.05169.

[3] P. A. Bernard, N Crampé, L. P. d’Andecy, L. Vinet, M. Zaimi,
Bivariate P-polynomial association schemes, to appear in Algebraic Combinatorics, (arXiv:2212.10824).

[4] N Crampé, L. Vinet, M. Zaimi, X. Zhang, A bivariate structure
for the non-binary Johnson scheme, J. Comb. Theory (A), 202:105829, 2024.

[5] N Crampé, M. Zaimi, Factorized $A_2$-Leonard pair, arXiv:2312.08312.