On the shape 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 the following conditions: (i) each of A,A∗ is diagonalizable; (ii) there exists an ordering {Vi} d i=0 of the eigenspaces of A such that A ∗Vi ⊆ Vi−1+Vi+Vi+1 for 0 ≤ i ≤ d, where V−1 = 0 and Vd+1 = 0; (iii) there exists an ordering {V ∗ i } δ i=0 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 and V ∗ δ+1 = 0; (iv) there is no subspace W of V such that AW ⊆ W , A ∗W ⊆ W , W 6= 0, W 6= V . We call such a pair a tridiagonal pair on V . It is known that d = δ and for 0 ≤ i ≤ d the dimensions of Vi, V ∗ i , Vd−i, V ∗ d−i coincide; we denote this common dimension by ρi. In this paper we prove that ρi ≤ ρ0 ( d i ) for 0 ≤ i ≤ d. It is already known that ρ0 = 1 if K is algebraically closed.