The finite unipotent groups consisting of bireflections
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 the corresponding modular invariant rings in a separate article.
scientific article; zbMATH DE number 7034988
scientific article