A vanishing theorem for Schur modules
type: Item
instance of: scholarly article
title: A vanishing theorem for Schur modules
zbMATH Open document ID: 0802.20037
published in: Journal of Algebra
publication date: 1994-12-14T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 589955
zbMATH Keywords: Schur algebra
zbMATH Keywords: finite dimensional \(k\)-algebra
zbMATH Keywords: polynomial representations
zbMATH Keywords: bases of Schur modules
zbMATH Keywords: skew partitions
zbMATH Keywords: induction
zbMATH Keywords: restriction
author: David J. Woodcock
reviewed by: Wilberd van der Kallen
MaRDI profile type: Publication
full work available at URL: jabr.1994.1126
OpenAlex ID: W2042332858
DOI: 10.1006/JABR.1994.1126
description: scientific article; zbMATH DE number 589955