Symplectic \(q\)-Schur algebras.
Schur algebras have been introduced by J. A. Green for general linear groups and by \textit{S. Donkin} [J. Algebra 104, 310-328 (1986; Zbl 0606.20038)] for reductive algebraic groups; these algebras have been used frequently in representation theory of algebraic groups. More recently, quantised versions of various Schur algebras have been defined and used, too. The article under review introduces quantised symplectic Schur algebras that for \(q=1\) specialise to Donkin's symplectic Schur algebras. The main results state that symplectic \(q\)-Schur algebras are cellular in the sense of Graham and Lehrer and integral quasi-hereditary (over a Laurent polynomial ring) in the sense of Cline, Parshall and Scott. A significant amount of complicated techniques is provided in order to prove these results. In particular, quantum symplectic bideterminants are introduced and the symplectic straightening formula the author had proved earlier [J. Algebra 244, No. 1, 19-44 (2001; Zbl 1018.16021)] is extended to the quantised setup. This article extends and improves the author's doctoral thesis of 1996 [Sitz.ber. Math.-Nat.wiss. Kl., Akad. Gem.nütz. Wiss. Erfurt 7, 187-196 (1996; Zbl 0901.20029)]. In the meantime, further work on (unquantised) symplectic Schur algebras has been done, in particular by \textit{G. Cliff} and \textit{A. Stokke} [J. Lond. Math. Soc., II. Ser. 72, No. 3, 585-608 (2005; Zbl 1134.20054)] and by Dipper, Doty and Hu (to appear).
scientific article; zbMATH DE number 5077825
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