Some algebra related to P-and Q-polynomial association schemes

Abstract

Inspired by the theory of P-and Q-polynomial association schemes we consider the following situation in linear algebra. Let F denote a field, and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A: V → V and A ∗ : V → V satisfying the following four conditions. (i) A and A ∗ are both diagonalizable on V. (ii) There exists an ordering V0,V1,...,Vd of the eigenspaces of A such that where V−1 = 0, Vd+1 = 0.