On the Gorenstein property for modular invariants
From the author's abstract: Let \(G\subset \mathrm{GL}(V)\) be a finite group, where \(V\) is a finite dimensional vector space over a field \(F\) of arbitrary characteristic. Let \(S(V)\) be the symmetric algebra of \(V\) and \(S(V)^G\) the ring of \(G\)-invariants. The author proves the followingNEWLINENEWLINETheorem: Suppose \(G\) contains no pseudo-reflection (of any kind). (1) If \(S(V)^G\) is Gorenstein, then \( G\subset GL(V)\). (2) If \(G\subset \mathrm{GL}(V)\), then the Cohen-Macaulay locus of \(S(V)^G\) coincides with its Gorenstein locus. In particular if \(S(V)^G\) is Cohen-Macaulay, then it is also Gorenstein.NEWLINENEWLINEThis extends well-known results of K. Watanabe in case \(\mathrm{gcd}(\mathrm{char}F, |G|) = 1\). It also confirms a special case of a conjecture due to G. Kemper, E. Kording, G. Malle, B.H. Matzat, D. Vogel and G. Wiese. A similar extension is given to D. Benson's theorem about the Gorenstein property of \((S(V)\otimes G(V))\), the polynomial tensor exterior algebra invariants. The author's proof uses non-commutative algebra methods in an essential way.
scientific article; zbMATH DE number 6022749
scientific article