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 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 of the eigenspaces of A∗ such that for 0⩽i⩽δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A∗W⊆W, W≠0, W≠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, , Vd-i, coincide; we denote this common dimension by ρi. In this paper we prove that for 0⩽i⩽d. It is already known that ρ0=1 if K is algebraically closed.