A Straightening Formula for Quantized Codeterminants

0. Introduction J.A. Green 6, x4] introduces codeterminants, which are products of pairs of natural basis elements of the Schur algebra S(n; r). These are shown to be parametrised by pairs of row-semistandard tableaux of shape , where 2 (n; r). If both the tableaux are standard, and the composition is dominant, then the codeterminant is said to be standard. The set of standard codeterminants forms a free Z-basis for the integral form of the classical Schur algebra. An algorithm for expressing an arbitrary codeterminant as a sum of standard ones has been described in 9]. It is natural to ask whether these results quantize. In this paper, we prove that they do. The Schur algebra has a quantum analogue, the so-called q-Schur algebra, S q (n; r), which is deened over the ring Zq; q ?1 ]. We will also be working with the v-Schur algebra, which is A S q (n; r), where A = Zv; v ?1 ] and v 2 = q. An algorithm to compute products in these algebras was established in 7]. We deene codeterminants for these algebras in an analogous way to Green 6], and after a careful analysis of certain products in S v (n; r), we