On polynomial invariant rings in modular invariant theory (Q6597188)

Let \(V\) be a finite-dimensional vector space over a field \(\Bbbk\). Let \(G\) be a group acting on \(V\) by \(\Bbbk\) linear transformations. Write \(S\) for the symmetric algebra \(\mathrm{Sym}\) \(V^*\) on the dual space \(V^*\) of \(V\). The \(G\) action on \(V\) induces a \(G\) action on \(S\). The ring of invariants is denoted by \(S^G\).\N\NAn area of invariant theory focuses on giving necessary and/or sufficient conditions under which \(S^G\) is a polynomial ring. If \(S^G\) is a polynomial ring, then \(S\) is a free \(S^G\)-module, so \(S^G\) is a direct summand of \(S\). The converse is not true in general, however an interesting case appears, when \(G\) is a finite \(p\)-group, and \(\Bbbk\) has characteristic \(p>0\). Then the Shank-Wehlau-Broer conjecture states that \(S^G\) is a polynomial ring if it is a direct summand of \(S\) as an \(S^G\)-module.\N\NThe present paper gives an affirmative answer for the Shank-Wehlau-Broer conjecture in two particular cases: first, when \(\Bbbk=\mathbb{F}_p\) and \(\mathrm{dim}_\Bbbk V=4\) (see the proof in Section 7) and second, when \(|G|=p^3\) (proof in Section 8). In order to prove these, it is shown that if \(\mathrm{dim}_\Bbbk V^G\geq\mathrm{dim}_\Bbbk V-2\), then the Hilbert ideal \(\mathcal{H}_{G,S}\) is a complete intersection (the proof can be found in Section 3). Another ingredient of the proofs uses the fact that a group \(G\) with order \(p^3\) has composition series formed by transvection groups (see Section 5 and 6 for details).