Abstract:
A q-analogue version of Schur algebras has been introduced by R. Dipper and G. James in connection with the representation theory of finite general linear groups. In recent work of Dipper, James and A. Matthas this theory has been extended to other groups of Lie type substituting Hecke algebras of type A by Ariki-Koike algebras leading to so called Cyclotomic q-Schur algebras. On the other hand the original q-Schur algebra can be obtained (up to Morita equivalence) using constructions from the theory of quantum groups. In this paper we apply these constructions to q-Schur algebras being related to the symplectic groups. These algebras appear in general as centralizer algebras of the Birman-Murakami-Wenzl algebras on tensor spaces. Setting the deformation parameter q to 1, we obtain classical symplectic Schur algebras in the sense of S. Donkin.
The main result in this paper is that symplectic q-Schur algebras are cellular in the sense of J. Graham and G. Lehrer and integrally quasi-hereditary as algebras over the ring of integer Laurent polynomials. A cellular basis is realized via a quantum symplectic version of bideterminants. While in the authors paper on Centralizer Coalgebras, FRT-Construction and Symplectic Monoids a symplectic version of the straightening formula for bideterminants has been presented in the classical case, we will give a quantized version of that formula, here.
03.02.2000, Sebastian Oehms, modifications 12.08.2004, 20.05.2022.