On finite unipotent transvection groups and their invariants (Q2922415)

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. Encyclopaedia of Mathematical Sciences 139. Invariant Theory and Algebraic Transformation Groups 8 (2011; Zbl 1216.14001)]). For arbitrary fields there is the paper [\textit{A. Broer}, Can. Math. Bull. 53, No. 3, 404--411, (2010; Zbl 1223.13005)] about abelian transvection groups.NEWLINENEWLINEThis paper is one more contribution to the classification of modular representations with polynomial invariant rings. Here \(G\) usually represents a \(p\)-group in characteristic \(p\), generated by tansvections. A transvection in this case is just a pseudo-reflection of order \(p\).NEWLINENEWLINEHere it is shown that the invariant ring for an extraspecial \(p\)-group is a polynomial algebra if and only if it is a tranvection group. Also, for a transvection \(p\)-group \(G\) of nilpotence class less or equal to \(2\), it is also proven that if the fixed point spaces \(V^G\) and \(\left(V^* \right)^G\) are one-dimensional, then \(\mathbb{F}[V]^G\) is polynomial ring. These theorems are presented in section \(6\) of this paper. In the first five sections the author studies the structure of transvection \(p\)-groups. For example, all the faithful transvection representations of extraspecial groups are described. Also of interest are the results about the commutator structure of tranvections.