The shape of a tridiagonal pair

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Let 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 A 3A ∗−[3]A 2A ∗A+[3]AA ∗A 2−A ∗A 3=0, A ∗3A−[3]A ∗2AA ∗+[3]A ∗AA ∗2−AA ∗3=0, where [3]=( q 3− q −3)/( q− q −1). Let ( ρ 0, ρ 1,…, ρ d ) denote the shape vector for A,A ∗ . We show the entries in this shape vector are bounded above by binomial coefficients as follows: ρ i⩽ d i (0⩽i⩽d). We obtain this result by displaying a spanning set for V.