The finite unipotent groups consisting of bireflections
type: Item
instance of: scholarly article
title: The finite unipotent groups consisting of bireflections
zbMATH Open document ID: 1442.13014
author: Juan-Miguel Gracia
published in: Journal of Group Theory
publication date: 2019-03-12T00:00:00Z
full work available at URL: Biref Accepted version.pdf
review text: Being able to find a connection between the invariant ring and the geometric structure of a finite group is a classical and important object of study in invaraint theory with a long history that can date back to the famous Shephard-Todd-Chevalley theorem which says that a nonmodular invariant ring \(F[W]^G\) is a polynomial algebra if and only if the action of \(G\) on \(W\) is generated by pseudo-reflections. This theorem usually fails to hold in the modular case, but Serre proved that one direction is feasible, i.e., if \(F[W]^G\) is polynomial, then \(G\) is a pseudo-reflection group. For the modular case, \textit{V. Kac} and \textit{K.-I. Watanabe} [Bull. Am. Math. Soc., New Ser. 6, 221--223 (1982; Zbl 0483.14002)] proved that if \(F[W]^G\) is a complete intersection, then \(G\) is a bireflection group; \textit{G. Kemper} [J. Algebra 215, No. 1, 330--351 (1999; Zbl 0934.13003)] proved that if \(F[W]^P\) is Cohen-Macaulay and \(P\) is a finite \(p\)-group, then \(P\) is a bireflection group. Motivated by Kemper's result [loc. cit.], an interesting question might be how to classify all finite \(p\)-groups generated by bireflections such that the corresponding invarint ring is Cohen-Macaulay. However, this question seems to be very hard and even impossible to solve completely. Thus one would expect special restriction conditions to reduce the range of groups. The paper under review considers a family of finite unipotent groups in which all elements are bireflections and such groups are called the \textit{pure bireflection groups}. The main purpose of the paper is to give a classification to pure bireflection groups and Theorem 1.5 is the main theorem which proves that if \(P\) is a finite \(p\)-group consisting of bireflections and \(p>2\), then \(P\) must belong to one of five specific pure bireflection groups: two-row groups; two-column groups; hook groups; and two kinds of exceptional groups. As the authors claim, these results of group theory serve to calculate…
reviewed by: Yin Chen
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 7034988
zbMATH Keywords: finite unipotent group
zbMATH Keywords: bi-reflection
MaRDI profile type: Publication
OpenAlex ID: W2890995308
cites work: Q4509240
cites work: Modular invariant theory
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DOI: 10.1515/JGTH-2018-0123
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description: scientific article; zbMATH DE number 7034988