Representations for Lie superalgebras of type C.
For the orthosymplectic Lie superalgebra \(C(n+1)=\text{osp}(2,2n)\), all finite-dimensional irreducible representations are known. A character formula for the typical representations has been given by \textit{V. G. Kac} [Differ. Geom. Meth. Math. Phys. II, Proc., Bonn 1977, Lect. Notes Math. 676, 597--626 (1978; Zbl 0388.17002)] and for the atypical representations by \textit{J. Van der Jeugt} [Commun. Algebra 19, 199--222 (1991; Zbl 0721.17004)]. In this paper, the author shows how the finite-dimensional irreducible representations \(V(\lambda)\) of \(C(n+1)\) can be labelled by a partition \(\lambda\) with \(\lambda_1\leq n\), and describes the character in terms of tableaux. Then he studies the tensor power \(T=\otimes^f V\) of the natural representation \(V\) and obtains an analogue of the Schur-Weyl duality in this case. In particular, it is shown that the centralizer algebra of \(C(n+1)\) on \(T\) is isomorphic to the Brauer algebra \(B_f(2-2n)\) for \(n>f\).
scientific article; zbMATH DE number 1868013