A classification of 2-simple prehomogeneous vector spaces of type I
type: Item
instance of: scholarly article
title: A classification of 2-simple prehomogeneous vector spaces of type I
zbMATH Open document ID: 0658.20026
DOI: 10.1016/0021-8693(88)90300-6
author: Tatsuo Kimura
author: Shin-ichi Kasai
author: Masaaki Inuzuka
author: Osami Yasukura
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: The purpose of this paper is the classification of 2-simple prehomogeneous vector spaces of type I. A prehomogeneous vector space (G,\(\rho\),V) is called a 2-simple prehomogeneous vector space when (1) \(G=GL(1)^{\ell}\times G_ 1\times G_ 2\) with simple algebraic groups \(G_ 1\) and \(G_ 2\), (2) \(\rho\) is the composition of a rational representation \(\rho '\) of \(G_ 1\times G_ 2\) of the form \[ \rho '=\rho_ 1\otimes \rho_ 1'+...+\rho_ k\otimes \rho_ k'+(\sigma_ 1+...+\sigma_ s)\otimes 1+1\otimes (\tau_ 1+...+\tau_{\ell}) \] with \(k+s+t=\ell\), where \(\rho_ i\), \(\sigma_ i\) (resp. \(\rho_ j'\), \(\tau_ j)\) are nontrivial irreducible representations of \(G_ 1\) (resp. \(G_ 2)\), and the scalar multiplications \(GL(1)^{\ell}\) on each irreducible component \(V_ i\) for \(i=1,...,\ell\) where \(V=V_ 1\oplus...\oplus V_{\ell}\). We say that a 2-simple prehomogeneous vector space (G,\(\rho\),V) is of type I if \(k\geq 1\) and at least one of \((GL(1)\times G_ 1\times G_ 2,\rho_ i\otimes \rho_ i')\) \((i=1,...,k)\) is a non-trivial prehomogeneous vector space. If \(k\geq 1\) and all \((GL(1)\times G_ 1\times G_ 2,\rho_ i\otimes \rho_ i')\) \((i=1,...,k)\) are trivial prehomogeneous vector spaces, it is called of type II. In a previous paper [\textit{T. Kimura}, \textit{S.-i. Kasai}, \textit{M. Taguchi} and \textit{M. Inuzuka}, Trans. Am. Math. Soc. 308, No.2, 433-494 (1988)], all 2-simple prehomogeneous vector spaces of type II have been classified. Together with the previous paper, this paper completes a classification of all 2-simple prehomogeneous vector spaces.
reviewed by: Masakazu Muro
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 4075392
zbMATH Keywords: 2-simple prehomogeneous vector spaces of type I
zbMATH Keywords: simple algebraic groups
zbMATH Keywords: rational representation
zbMATH Keywords: irreducible representations
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90300-6
OpenAlex ID: W2059986513
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description: scientific article; zbMATH DE number 4075392