Rings of invariants for the three-dimensional modular representations of elementary abelian \(p\)-groups of rank four (Q2631159)

This article continues a program of research started by Campbell, Shank and Wehlau [\textit{H. E. A. Campbell} et al., Transform. Groups 18, No. 1, 1--22 (2013; Zbl 1264.13009)]: the computation of rings of modular invariants for elementary abelian \(p\)-groups acting linearly on vector spaces of small dimension. The three-dimensional representations of these groups were classified by socle series in [loc. cit.], the only case in which the rings of invariants are not known is for those of type \((1,1,1)\), i.e. when \(\dim(V^G)=1\) and \(\dim((V/V^G)^G)=1\). For this socle series the rings of invariants of elementary abelian \(p\)-groups of rank 3 were studied in [loc. cit.]. In the present article the authors study groups of rank four. The methods used are essentially the same: representations are parameterised by orbits of the action of GL\(_4(\mathbb{F}_p)\) on two-by-four matrices and the SAGBI/divide-by-\(x\) algorithm from [loc. cit.] used to compute a SAGBI basis. This gives an explicit generating set for each ring of invariants considered, as well as the number of independent relations. The authors prove that the rings of modular invariants for elementary abelian \(p\)-groups of rank for acting on a vector space of dimension three are complete intersections with embedding dimension at most five, confirming a conjecture of Campbell, Shank and Wehlau.