The split decomposition of a tridiagonal pair

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i)–(iv) below: (i) Each of A, A∗ is diagonalizable. (ii) There exists an ordering V0, V1, . . . , Vd of the eigenspaces of A such that A ∗Vi ⊆ Vi−1 + Vi + Vi+1 for 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 for 0 ≤ i ≤ δ, where V ∗ −1 = 0, V ∗ δ+1 = 0. (iv) There is no subspace W of V such that both AW ⊆ W , A∗W ⊆ W , other than W = 0 and W = V . We call such a pair a tridiagonal pair on V . In this note we obtain two results. First, we show that each of A,A∗ is determined up to affine transformation by the Vi and V ∗ i . Secondly, we characterize the case in which the Vi and V ∗ i all have dimension one. We prove both results using a certain decomposition of V called the split decomposition.