Abstract:
Using a construction by Faddeev, Reshetikhin and Takhtadjian (1990) together with some additional relations we obtain a quantization of the coordinate ring of the symplectic monoid. The latter is defined as the Zariski-closure of the group of symplectic similitudes inside the monoid of n by n matrices. It has been considered by Grigor'ev (1981) and Doty (1996), too. We define symplectic q-Schur algebras as the dual algebras to the homogeneous summands of these quantized coordinate rings. In the classical specialization one obtains the symplectic Schur algebras defined by Donkin (1987). We define a quantum symplectic version of bideterminants and prove an analogue to the well known straightening formula. This yields cellular bases (in the sense of Graham and Lehrer) for these q-Schur algebras. Furthermore the criterion for quasi-heriditaryness of a cellular algebra is checked to hold. It is easy to see, that the symplectic q-Schur algebras are the centralizer algebras of the Birman-Murakami-Wenzl algebras in the generic case (over the rational function field in q). We show that the same is true for other specializations, at least if q is not a certain root of unity. Finally we are able to improve some results of Doty concerning the classical symplectic Schur algebras over a field of characteristic zero, to fields of arbitrary characteristic.
Remark:
A brief survey of the main results of the thesis can be found in the manusscript of a talk I held in Jena in May 1996. As the manusscript of the thesis it is in German. I started to produce two preprints of the material in English. For two years I hadn't got any time to continue working on them. But now the first one is ready for submission, whereas the second one is still under correction. Nevertheless you may download postscriptfiles of these two manuscripts:
The first paper contains material I presented at a workshop in Oberflockenbach . In addition it contains a proof of the straightening formular for symplectic monoids in the classical case. The procedure is different from that given in the thesis (adapted to this easier case and thus much clearer).
The second paper contains a proof of the straightening formular for quantum symplectic monoids. The procedure is as well different from that given in the thesis. It is more like the treatment in the first paper. The aim of the paper is the main result of the thesis, that symplectic q-Schur-algebras a cellular and quasi-heriditary.
04.06.1997, Sebastian Oehms, modifications: 04.02.2000, 20.05.2022.