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

TL;DR

This paper demonstrates that the non-binary Johnson scheme possesses a bivariate $Q$-polynomial structure, enriching its algebraic properties and linking it to families of bivariate orthogonal polynomials.

Contribution

It establishes the bivariate $Q$-polynomial property for the non-binary Johnson scheme, complementing its known $P$-polynomial structure and exploring associated algebraic frameworks.

Findings

The non-binary Johnson scheme is a bivariate $Q$-polynomial association scheme.

The scheme exhibits bispectral properties related to bivariate orthogonal polynomials.

The algebra of bispectral operators and subconstituent algebra are analyzed.

Abstract

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.