A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplications
type: Item
instance of: scholarly article
title: A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplications
zbMATH Open document ID: 0533.14024
DOI: 10.1016/0021-8693(83)90138-2
author: Tatsuo Kimura
published in: Journal of Algebra
publication date: 1983-01-01T00:00:00Z
review text: The purpose of this paper is the complete classification of the complex prehomogeneous vector spaces such that the group is reductive with scalar multiplications. - Let G be a connected complex linear algebraic group and let \(\rho\) be a finite dimensional rational representation on the complex vector space V. We say that the triplet \((G,\rho\),V) is a prehomogeneous vector space if V contains a Zariski-dense G-orbit. Especially, we suppose that \(G=GL(1)^{\ell}\times G_ 1\times...\times G_ k\) with \(G_ i\) simple linear algebraic groups, and \(V=V_ 1\oplus...\oplus V_{\ell}\) where the restriction of \(\rho\) on \(V_ j\) is an irreducible representation of \(G_ 1\times...\times G_ k\). The author has already classified all the prehomogeneous vector spaces when \(\ell =1\), i.e., irreducible representations, in the joint work with \textit{M. Sato} [Nagoya Math. J. 65, 1-155 (1977; Zbl 0321.14030)]. In this paper, the author solves the problem of classification when \(\ell \geq 2\). In the last section of this paper, a complete table of prehomogeneous vector spaces and their relative invariants is given.
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 3845752
zbMATH Keywords: classification of complex prehomogeneous vector spaces
reviewed by: Masakazu Muro
MaRDI profile type: Publication
full work available at URL: 0021-8693(83)90138-2
OpenAlex ID: W1977900194
cites work: A classification of irreducible prehomogeneous vector spaces and their relative invariants
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description: scientific article; zbMATH DE number 3845752