Bivariate $P$-polynomial association schemes

Abstract

Bivariate P-polynomial association scheme of type $(\hat{I}\pm ,{\hat{I}}^2)$ are defined as a generalization of the P-polynomial association schemes. This generalization is shown to be equivalent to a set of conditions on the intersection parameters. A number of known higher rank association schemes are seen to belong to this broad class. Bivariate Q-polynomial association schemes are similarly defined.