The main goal of the paper under review is the construction of a cellular basis for the symplectic \(q\)-Schur algebra. The author begins by summarizing the definition of the ordinary Schur algebra due to J. A. Green [“Polynomial representations of \(\text{GL}_n\)”, Lect. Notes Math. 830 (1980; Zbl 0451.20037)] and its \(q\)-analogue due to R. Dipper and G. James [Proc. Lond. Math. Soc., III. Ser. 59, No. 1, 23-50 (1989; Zbl 0711.20007)]. Then he considers the symplectic monoid which was already studied by D. Yu. Grigor’ev [Dokl. Akad. Nauk SSSR 257, 1040-1044 (1981; Zbl 0479.22003)], and the ordinary symplectic Schur algebra introduced by S. Donkin [Proc. Symp. Pure Math. 47, 69-80 (1987; Zbl 0648.20048)] and recently investigated by S. Doty [J. Pure Appl. Algebra 123, No. 1-3, 165-199 (1998)]. After that he recalls bideterminants and provides the classical straightening formula whose dual was used by J. A. Green [J. Pure Appl. Algebra 88, No. 1-3, 89-106 (1993; Zbl 0794.20053)] to develop in elegant approach to the quasi-hereditarity of the Schur algebra. The author also explains the concept of a cellular basis and a cellular algebra introduced by J. J. Graham and G. I. Lehrer [Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)] to deal in a uniform way with the representation theory of algebras similar to the Schur algebra (and thus generalizing the ideas of Green) together with the dual concept of a cellular coalgebra.
In the main part of the paper the author applies a slight modification of the Faddeev-Reshetikhin-Takhtadzhyan construction [N. Yu. Reshetikhin, L. A. Takhtadzhyan and L. D. Faddeev, Leningr. J. Math. 1, 193-225 (1990); translation from Algebra Anal. 1, No. 1, 178-206 (1989; Zbl 0715.17015)] to construct a \(q\)-analogue of the coordinate algebra of the symplectic monoid over an arbitrary commutative ring in which \(q\) is a unit. This is a graded bialgebra whose homogeneous components are finite-dimensional subcoalgebras. Therefore the dual algebra of a homogeneous component can be used to define a symplectic \(q\)-Schur algebra. After introducing a \(q\)-version of symplectic bideterminants, the author describes a \(q\)-analogue of the classical straightening formula from which he obtains (under a certain restriction on \(q\)) a cellular basis for the symplectic \(q\)-Schur algebra. This enables him (under the same restriction on \(q\) as before) to identify the symplectic \(q\)-Schur algebra with a suitable centralizer of the Birman-Murakami-Wenzl algebra and to extend some other results of Doty for \(q=1\) from an algebraically closed field of characteristic zero to an algebraically closed field of arbitrary characteristic.
The paper is very well written and gives a nice first introduction into the theory of Schur algebras and their \(q\)-analogues. There are no proofs; but these can be found in the author’s doctoral thesis [Symplektische \(q\)-Schur-Algebren, Berichte aus der Algebra, Shaker-Verlag (1997)], where he also verifies that the \(q\)-Schur algebra is quasi-hereditary.