Minimal relative Hilbert-Kunz multiplicity (Q1425779)

Let \(V\) be an \(n\)-dimensional vector space over a field and \(S = S(V)\) the symmetric algebra. For a subgroup \(G\) of \(\mathrm{GL}(V)\) let \(S^G\) denote the ring of invariants. In general there is a big difference between the modular and the non-modular situation in invariant theory. In the case of \(L\) a finite non-modular group whose ring of invariants form a polynomial ring and \(H\) a subgroup of \(L\) containing the derived group of \(L\) there are necessary and sufficient conditions on \(H\) for its invariant ring \(S^H\) to be a hypersurface. In fact, \textit{H. Nakajima} [J. Algebra 80, 279--294 (1983; Zbl 0524.14013)] has shown that if \(S^H\) is a hypersurface, then between \(H\) and \(L\) there is a group \(G\) with polynomial invariant ring such that \(S^H = S^G[b]\) for some \(b \in S^H.\) The modular analog of this result is not true. In the present paper the authors extend this result to the modular case. In particular, for \(G\) a finite modular \(p\)-group over F\(_p\) with polynomial invariant ring and \(H\) a subgroup of \(G\) containing the derived group of \(G\) there are necessary and sufficient conditions on \(H\) to guarantee that \(S^H = S^G[b].\) In their investigations the authors prove a generalization of a result of \textit{H. E. A. Campbell} and \textit{I. P. Hughes} [J. Algebra 211, 549--561 (1999; Zbl 0926.13003)].