The shape of a tridiagonal pair

Let denote an algebraically closed field with characteristic 0. Let V denote a vector space over with finite positive dimension and let A,A* denote a tridiagonal pair on V. We make an assumption about this pair. Let q denote a nonzero scalar in that is not a root of unity. We assume A and A* satisfy the q-Serre relations A3A*−[3]A2A*A+[3]AA*A2−A*A3=0,A*3A−[3]A*2AA*+[3]A*AA*2−AA*3=0,where [3]=(q3−q−3)/(q−q−1). Let (ρ0,ρ1,…,ρd) denote the shape vector for A,A*. We show the entries in this shape vector are bounded above by binomial coefficients as follows: We obtain this result by displaying a spanning set for V.