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

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 AVi ⊆ Vi−1 + Vi + Vi+1 (0 ≤ i ≤ d), where V−1 = 0, Vd+1 = 0. (iii) There exists an ordering V ∗ 0 , V ∗ 1 , . . . , V ∗ δ of the eigenspaces of A ∗ such that AV ∗ i ⊆ V ∗ i−1 + V ∗ i + V ∗ i+1 (0 ≤ i ≤ δ), where V ∗ −1 = 0, V ∗ δ+1 = 0. (iv) There is no subspace W of V such that both AW ⊆ W , AW ⊆ W , other than W = 0 and W = V . We call such a pair a TD pair. Referring to the above TD pair, we show d = δ. We show that for 0 ≤ i ≤ d, the eigenspaces Vi and V ∗ i have the same dimension. Denoting this common dimension by ρi, we show the sequence ρ0, ρ1, . . . , ρd is symmetric and unimodal, i.e. ρi−1 ≤ ρi for 1 ≤ i ≤ d/2 and ρi = ρd−i for 0 ≤ i ≤ d. We show that there exists a sequence of scalars β, γ, γ, ̺, ̺ taken from F such that both 0 = [A,AA − βAAA+AA − γ(AA +AA)− ̺A], 0 = [A, AA− βAAA +AA − γ(AA+AA)− ̺A], where [r, s] means rs − sr. The sequence is unique if d ≥ 3. Let θi (resp. θ ∗ i ) denote the eigenvalue of A (resp. A) associated with Vi (resp. V ∗ i ), for 0 ≤ i ≤ d. We show the expressions θi−2 − θi+1 θi−1 − θi , θ i−2 − θ ∗ i+1 θ i−1 − θ ∗ i both equal β + 1, for 2 ≤ i ≤ d − 1. We hope these results will ultimately lead to a complete classification of the TD pairs.