The shape of a tridiagonal pair

AbstractLet K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K 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 K 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:ρi⩽di(0⩽i⩽d).We obtain this result by displaying a spanning set for V