A classification of a certain class of reductive prehomogeneous vector spaces. II
type: Item
instance of: scholarly article
title: A classification of a certain class of reductive prehomogeneous vector spaces. II
zbMATH Open document ID: 0688.20024
DOI: 10.1016/0021-8693(90)90243-H
author: Shin-ichi Kasai
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: [Part I cf. Commun. Algebra 17, No.6, 1425-1447 (1989; Zbl 0671.20037).] The purpose of this paper is to classify all reductive prehomogeneous vector spaces \((\tilde G,{\tilde\rho},\tilde V)\) when each irreducible component \((GL(1)\times G,\Lambda_ 1\otimes \rho_ i,V_ i)\) is castling equivalent to a non-trivial reduced irreducible prehomogeneous vector space. Let G be a connected semi-simple linear algebraic group. Every finite dimensional rational representation \(\rho\) : \(G\to GL(V)\) of G can be written as a direct sum \(\rho =\rho_ 1\oplus...\oplus \rho_ k\) of irreducible rational representations \(\rho_ i: G\to GL(V_ i)\) of G in \(V_ i\) \((i=1,...,k)\). Let \({\tilde \rho}\) be the action of \(\tilde G=GL(1)^ k\times G\) on \(\tilde V=V_ 1\oplus...\oplus V_ k\) which is the composition of \(\rho =\rho_ 1\oplus...\oplus \rho_ k\) and independent scalar multiplications \(GL(1)^ k\) on each irreducible component \(V_ i\). We call such a triplet \((\tilde G,{\tilde\rho},\tilde V)\) a reductive prehomogeneous vector space if it is aprehomogeneous vector space. All the reductive prehomogeneous vector spaces are classified when each irreducible component \((GL(1)\times G,\Lambda \otimes \rho_ i,V_ i)\) \((i=1,...,k)\) is a regular prehomogeneous vector space.
reviewed by: Masakazu Muro
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 4127501
zbMATH Keywords: reductive prehomogeneous vector spaces
zbMATH Keywords: irreducible prehomogeneous vector space
zbMATH Keywords: connected semi-simple linear algebraic group
zbMATH Keywords: irreducible rational representations
zbMATH Keywords: action
MaRDI profile type: Publication
cites work: Q4206873
cites work: A classification of a certain class of reductive prehomogeneous vector spaces
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cites work: Some P.V.-Equivalences and a Classification of 2-Simple Prehomogeneous Vector Spaces of Type II
cites work: A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplications
cites work: Q3483468
cites work: A classification of irreducible prehomogeneous vector spaces and their relative invariants
cites work: Construction de certaines sous-algèbres remarquables dans les algèbres de Lie semi-simples
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description: scientific article; zbMATH DE number 4127501