Codeterminants and q-Schur Algebras

We study codeterminants in the q-Schur algebra S (q) (n,r) and prove that the standard ones form a basis of S (q) (n,r), using a quantized version of the D,sarm,nien matrix. We find elements of the form F (S) 1(lambda) E (T) in Lusztig's modified enveloping algebra of gl(n), which, up to powers of q, map to the basis of standard codeterminants, where F (S) aU (-) and E (T) aU (+) are explicitly given products of root vectors, depending on Young tableaux S and T.