A canonical filtration for certain rational modules
type: Item
instance of: scholarly article
title: A canonical filtration for certain rational modules
zbMATH Open document ID: 0535.20022
DOI: 10.1007/BF01159188
author: Eric M. Friedlander
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173551
review text: Let G be a simple, simply connected algebraic group and let \(B\subset G\) be a Borel subgroup. A filtration of a rational G-module is called a ''good filtration'' if its successive quotients are each obtained by induction from one-dimensional B-modules. The author gives a cohomological characterization of rational G-modules of countable dimension which admit a good filtration; this extends a result of \textit{S. Donkin} [Math. Z. 177, 1-8 (1981; Zbl 0455.20029)] for finite dimensional G-modules. Moreover, the author demonstrates that a rational G-module of arbitrary dimension satisfying Donkin's cohomological condition admits a ''canonical filtration'' associated to a natural partial ordering of induced modules. The usefulness of the author's criterion is demonstrated by a theorem providing a calculation in all degrees of the rational cohomology of a rational G-module obtained by a single Frobenius twist from one with a canonical filtration.
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 3849485
zbMATH Keywords: rational modules
zbMATH Keywords: simple, simply connected algebraic group
zbMATH Keywords: Borel subgroup
zbMATH Keywords: induction
zbMATH Keywords: good filtration
zbMATH Keywords: rational cohomology
zbMATH Keywords: Frobenius twist
MaRDI profile type: Publication
cites work: Cohomology, hyperalgebras, and representations
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cites work: Cohomology of algebraic linear groups
cites work: Sheaf cohomology on G/B and tensor products of Weyl modules
description: scientific article; zbMATH DE number 3849485