Bivariate Q-polynomial structures for the nonbinary Johnson scheme and the association scheme obtained from attenuated spaces

The study of P-polynomial association schemes (distanceregular graphs) and Q-polynomial association schemes, and in particular P- and Q-polynomial association schemes, has been a central theme not only in the theory of association schemes but also in the whole study of algebraic combinatorics in general. Leonard's theorem (1982) says that the spherical functions (or the character tables) of P- and Q-polynomial association schemes are described by Askey -Wilson orthogonal polynomials or their relatives. These polynomials are onevariable orthogonal polynomials. It seems that the new attempt to define and study higher rank P- and Q-polynomial association schemes had been hoped for, but had gotten only limited success. The first very successful attempt was initiated recently by Bernard-Cramp & eacute;-d'Andecy-Vinet-Zaimi, and then followed by Bannai-Kurihara-Zhao-Zhu. The general theory and some explicit examples of families of higher rank (multivariate) P- and/or Q-polynomial association schemes have been obtained there. The main purpose of the present paper is to prove that some important families of association schemes are shown to be bivariate Q-polynomial. Namely, we show that all the nonbinary Johnson association schemes and all the association schemes obtained from attenuated spaces are bivariate Q -polynomial. It should be noted that the parameter restrictions needed in the previous papers are completely lifted in this paper. Our proofs are done by explicitly calculating the Krein parameters of these association schemes. At the end, we mention some speculations and indications of what we can expect in the future study. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).