A vanishing theorem for Schur modules
Recall that the Schur algebra \(S_ d(G)\) of \(G = GL_ n(k)\) is a finite dimensional \(k\)-algebra whose representations are the polynomial representations of \(G\) which are homogeneous of degree \(d\). The author shows that ``standard tabloids'' yield bases of Schur modules of \(S_ d(G)\) for a class of shapes containing all skew partitions. He extends a short exact sequence of \textit{K. Akin} and \textit{D. A. Buchsbaum} [Adv. Math. 72, 171-210 (1988; Zbl 0681.20028)] to his class and derives cohomological consequences for induction from -- or restriction to -- modules over the Schur algebra \(S_ d(B)\) of the subgroup \(B\) of lower triangular matrices. The results are analogous to \textit{G. Kempf}'s vanishing theorem [Ann. Math., II. Ser. 103, 557-591 (1976; Zbl 0327.14016)] and some of its representation theoretic consequences.
scientific article; zbMATH DE number 589955