2 00 3 The shape of a tridiagonal pair ∗

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 which is not a root of unity. We assume A and A satisfy the q-Serre relations AA − [3]AAA+ [3]AAA −AA = 0, AA− [3]AAA + [3]AAA −AA = 0, where [3] = (q−q)/(q−q). 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 ≤ ( d i ) (0 ≤ i ≤ d). We obtain this result by displaying a spanning set for V .