Bivariate $P$- and $Q$-polynomial structures of the association schemes based on attenuated spaces

TL;DR

This paper investigates the bivariate $P$- and $Q$-polynomial structures of association schemes based on attenuated spaces, revealing their bispectral properties and algebraic structures using special polynomial relations.

Contribution

It introduces the bispectral properties of these association schemes and explores their algebraic structures, connecting them to known polynomial families and non-binary Johnson schemes.

Findings

Derived recurrence and difference relations for eigenvalue polynomials.

Identified bispectral algebra and subconstituent algebra of the schemes.

Compared properties with non-binary Johnson schemes through a limiting process.

Abstract

The bivariate $P$- and $Q$-polynomial structures of association schemes based on attenuated spaces are examined using recurrence and difference relations of the bivariate polynomials which form the eigenvalues of the scheme. These bispectral properties are obtained from contiguity relations of univariate dual $q$-Hahn and affine $q$-Krawtchouk polynomials. The bispectral algebra associated to the bivariate polynomials is investigated, as well as the subconstituent algebra of the schemes. The properties of the schemes are compared to those of the non-binary Johnson schemes through a limit.