Symplectic \(q\)-Schur algebras.
type: Item
instance of: scholarly article
title: Symplectic \(q\)-Schur algebras.
zbMATH Open document ID: 1118.20047
author: Sebastian Oehms
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 5077825
zbMATH Keywords: quantised Schur algebras
zbMATH Keywords: symplectic Schur algebras
zbMATH Keywords: quantum symplectic bideterminants
zbMATH Keywords: symplectic straightening formula
reviewed by: Steffen Koenig
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2005.07.030
OpenAlex ID: W2006415709
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DOI: 10.1016/J.JALGEBRA.2005.07.030
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description: scientific article; zbMATH DE number 5077825