On a certain class of prehomogeneous vector spaces
type: Item
instance of: scholarly article
title: On a certain class of prehomogeneous vector spaces
zbMATH Open document ID: 0633.14030
DOI: 10.1016/0022-4049(87)90051-X
author: Jun-ichi Jgusa
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let k be a field of characteristic 0. Let \(X=Aff^ n\) be an n- dimensional k-affine space and let \(S:=\{x\in X\); \(f(x)=0\}\) be an irreducible hypersurface in X defined over k. We denote \(Y:=X-S\). The author presents the following problem: classify all the pairs (G,Y) where G is a connected irreducible k-subgroup of GL(X) which acts on Y transitively. Such pairs become special cases of prehomogeneous vector spaces defined over k. He solves this problem in the case when G splits over k and a quaternion division k-algebra exists. In his proof, he utilizes effectively the property that the roots of b-function of \(f^ s\) are all integers. This property is held under ``castling transform'', hence it is sufficient to classify all the ``reduced'' pairs in the sense of Sato-Kimura.
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 4029751
zbMATH Keywords: action of linear group
zbMATH Keywords: prehomogeneous vector spaces
zbMATH Keywords: roots of b- function
zbMATH Keywords: castling transform
reviewed by: Masakazu Muro
MaRDI profile type: Publication
cites work: Q5522255
cites work: Q4935949
cites work: Q5830680
cites work: Spinors of 13-dimensional space
cites work: A Classification of Spinors Up to Dimension Twelve
cites work: Q3215424
cites work: On functional equations of complex powers
cites work: Q5622399
cites work: Theb-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces
cites work: On the microlocal structure of the regular prehomogeneous vector space associated with \(SL(5)\times GL(4)\). I
cites work: Q3885923
cites work: A classification of irreducible prehomogeneous vector spaces and their relative invariants
cites work: On zeta functions associated with prehomogeneous vector spaces
cites work: Galois Cohomology
full work available at URL: 0022-4049(87)90051-x
OpenAlex ID: W2078374665
description: scientific article; zbMATH DE number 4029751