, talk at the Darstellungstheorietage Jena, 16.-18.5.1996
Appeared:
B. Külshammer, K. Rosenbaum (ed.): Darstellungstheorietage Jena 1996. Akad. gemein. Wiss. Erfurt, Sitzungsber. Math.-Nat. Klasse 7, 1996, p. 187-196.
Abstract:
This is a brief survey of my doctoral thesis. Let R be the ring of integer Laurent polynomials in an indeterminant q. The Birman-Murakami-Wenzl algebra over R acts on the r-fold tensor product of the natural module of the quantum symplectic group centralizing the action of the latter one. It turns out that the centralizing algebra of this action is a q-analogue version of S. Donkin's (1987) resp. S. Doty's (1996) symplectic Schur algebra and is therefore considered as symplectic q-Schur algebra. Furthermore this algebra has a free R-basis which is cellular in the sense of J. Graham and G. Lehrer (1996). If S is a commutative R-algebra, then in general tensoring the symplectic q-Schur algebra by S yields the centralizing algebra of the Birman-Murakami-Wenzl algebra over S. But for certain specializations corresponding to q-analogues of integers this is still an unsolved problem. Finally applications to the classical case are given.
Remark:
The present German paper doesn't contain any proof. For an English manuscript containing the proofs click here .