On finite unipotent transvection groups and their invariants
type: Item
instance of: scholarly article
zbMATH Open document ID: 1302.13009
author: Peter Fleischmann
published in: The Quarterly Journal of Mathematics
publication date: 2014-10-10T00:00:00Z
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20F12
zbMATH DE Number: 6353629
zbMATH Keywords: modular invariant theory
zbMATH Keywords: invariant rings
zbMATH Keywords: unipotent transvection groups
zbMATH Keywords: extraspecial groups
MaRDI profile type: Publication
full work available at URL: has042
OpenAlex ID: W1966983945
DOI: 10.1093/QMATH/HAS042
title: On finite unipotent transvection groups and their invariants
review text: Let \(G\) be a finite group with a dimensional representation \(V\) over a field \(\mathbb{F}\). Then \(G\) acts naturally on the symmetric algebra \(S:=\mathbb{F}[V]:=\text{Sym}(V^*)\) by graded algebra automorphisms. One of the main problems of invariant theory is the investigation of the structure of the \textit{ring of invariants} NEWLINE\[NEWLINE\mathbb{F}[V]^G:=\{f\in \mathbb{F}[V]\;|\;g\cdot f=f,\;\forall\;g\in G\}.NEWLINE\]NEWLINE When the characteristic of \(\mathbb{F}\) divides the order of \(G\) we speak of \textit{modular invariant theory}.NEWLINENEWLINEOne important open question in modular invariant theory is the classification of the linear representations of \(G\) for which the invariant ring is polynomial. It is known by a result of Serre that if \(\mathbb{F}[V]^G\) is a polynomial algebra then the linear representation of \(G\) is generated by pseudo-reflections. However, the converse is known not to be true in the modular case. In the nonmodular case (when the characteristic of \(\mathbb{F}\) does not divide the order of \(G\)), a theorem of Sheppard-Todd-Chevalley and another of Serre show that the converse also holds. For more details see [\textit{H. E. A. E. Campbell} and \textit{D. L. Wehlau}, Modular invariant theory. Encyclopaedia of Mathematical Sciences 139. Invariant Theory and Algebraic Transformation Groups 8 (2011; Zbl 1216.14001)].NEWLINENEWLINEIn the modular case, the classification of the linear representations of \(G\) for which the invariant ring is polynomial is not yet complete. Some results are known. In the paper [\textit{G. Kemper} and \textit{G. Malle}, Transform. Groups 2, No. 1, 57--89 (1997; Zbl 0899.13004)] all the irreducible linear representations with a polynomial ring of invariants are determined. Nakajima gave a precise criterion for unipotent groups defined over the prime field to have polynomial ring of invariants (see chapter 8 of \textit{H. E. A. E. Campbell} and \textit{D. L. Wehlau} [Modular invariant theory. …
reviewed by: Jorge Nélio Ferreira
description: scientific article; zbMATH DE number 6353629