Some prehomogeneous representations defined by cubic forms
type: Item
instance of: scholarly article
title: Some prehomogeneous representations defined by cubic forms
zbMATH Open document ID: 0920.17015
author: Hisatoshi Ikai
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1999-05-05T00:00:00Z
review text: There are four fundamental irreducible prehomogeneous vector spaces with relative invariants of degree 4 in the classification table of Sato-Kimura of reduced prehomogeneous vector spaces. In this paper, the author constructs ``\(k\)-forms'' of these prehomogeneous vector spaces and determines the generic stabilizer where \(k\) is an arbitrary commutative ring. Instead of considering in the framework of classical theory of prehomogeneous vector space, he starts with the axiomatic construction of the linear representation of an algebraic group \(G(k)\) defined over the commutative ring \(k\). Then he determines the stabilizer subgroup of a generic point belonging to an open subscheme defined by a quartic form. Under some conditions, by ``extending'' to the algebraic closed field \(K\) from the commutative ring \(k\), it is proved that the group \(G(K)\) acts on the open subscheme transitively.
reviewed by: Masakazu Muro
Mathematics Subject Classification ID: 17C30
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 1268107
zbMATH Keywords: prehomogeneous vector spaces
zbMATH Keywords: generic stabilizer
zbMATH Keywords: algebraic group
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178224935
OpenAlex ID: W2120320314
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DOI: 10.2748/TMJ/1178224935
description: scientific article; zbMATH DE number 1268107