Homomorphisms between neighbouring Weyl modules
type: Item
instance of: scholarly article
title: Homomorphisms between neighbouring Weyl modules
zbMATH Open document ID: 0603.20040
DOI: 10.1016/0021-8693(86)90189-4
author: M. Koppinen
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let G be a simply connected semi-simple algebraic group over an algebraically closed field k of positive characteristic, defined and split over the prime field of k. In this paper, the author deals with homomorphisms between the Weyl modules of G. He introduces for the sets of weights the concept of small dominant intervals and shows that if the dominant weights \(\lambda\) and \(\chi\) belong to the same small dominant interval, then there is a non-zero homomorphism \(f: V(\lambda)\to V(\chi)\) between corresponding Weyl modules. Furthermore, he shows that for a special class of small dominant intervals, f is unique up to a scalar multiple and the composite map of two such homomorphisms is non- zero. To deal with the Weyl modules the author also introduces in the set of weights a new partial order called the regular linkage relation. On dominant weights it implies the relation used by \textit{J. C. Jantzen} [Moduln mit einem höchsten Gewicht (Lect. Notes Math. 750, 1979; Zbl 0426.17001)], while for dominant weights in alcoves the relations are equivalent. In various results, including the strong linkage principle [cf. \textit{H. H. Andersen}, J. Reine Angew. Math. 315, 53-59 (1980; Zbl 0439.20026)], the author shows that the new relation can substitute for the above without any significant changes in the proofs and leads to slight improvements of some older theorems.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 3973164
zbMATH Keywords: simply connected semi-simple algebraic group
zbMATH Keywords: homomorphisms
zbMATH Keywords: Weyl modules
zbMATH Keywords: small dominant intervals
zbMATH Keywords: dominant weights
zbMATH Keywords: regular linkage relation
zbMATH Keywords: strong linkage principle
reviewed by: Eiichi Abe
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90189-4
OpenAlex ID: W2090881290
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cites work: Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne.
cites work: On the composition factors of Weyl modules.
cites work: On the translation functors for a semisimple algebraic group.
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description: scientific article; zbMATH DE number 3973164