The shape of a tridiagonal pair

Abstract

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,B 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 B satisfy the q-Serre relations (i) A^3B - [3]A^2BA + [3]ABA^2 - BA^3=0; (ii) B^3A - [3]B^2AB + [3]BAB^2 - AB^3=0, where [3]=(q^3-q^{-3})/(q-q^{-1}). Let (\rho_0, \rho_1,...,\rho_d) denote the shape vector for A,B. We show the entries in this shape vector are bounded above by binomial coefficients. Indeed we show \rho_i is at most (d \atop i) for 0 \leq i \leq d. We obtain this result by displaying a spanning set for V.