On modular invariants of a vector and a covector
Let \(\mathbb{F}_q\) denote the finite field with \(q=p^r\) elements, and \(G\) be a subgroup of \(\mathrm{GL}_2(\mathbb{F}_q)\). Let \(V\) be the natural 2-dimensional representation of \(G\). The author considers the ring of invariants \(\mathbb{F}_q[V \oplus V^*]^G\) in the cases where \(G\) is \(U_2(\mathbb{F}_q)\) (the group of unipotent upper triangular matrices over \(\mathbb{F}_q\)), \(\mathrm{SL}_2(\mathbb{F}_q)\) or \(\mathrm{GL}_2(\mathbb{F}_q)\). In the case where \(G=\mathrm{SL}_2(\mathbb{F}_q)\), \(V\) is self-dual and we have that \(\mathbb{F}_q[V \oplus V^*]^G\) is isomorphic to \(\mathbb{F}_q[V \oplus V]^G\). Algebra generators for this ring of invariants were computed by Campbell and Hughes in the case \(q=p\) [\textit{H. E. A. Campbell} and \textit{I. P. Hughes}, J. Pure Appl. Algebra 112, No. 1, 1--12 (1996; Zbl 0872.13006)]. Using similar methods, the author computes a generating set for arbitrary \(q\). It is observed that one obtains a Gorenstein algebra. The author then uses the relative trace map to compute algebra generators for \(\mathbb{F}_q[V \oplus V^*]^G\) where \(G=\mathrm{GL}_2(\mathbb{F}_q)\). Again it is observed that this is a Gorenstein algebra. This confirms a special case of conjecture of \textit{C. Bonnafé} and \textit{G. Kemper} [J. Algebra 335, No. 1, 96--112 (2011; Zbl 1267.13010)], who showed that for arbitrary \(n\) and \(q\) the rings of invariants \(\mathbb{F}_q[V \oplus V^*]^{\mathrm{GL}_n(\mathbb{F}_q)}\) (where \(V\) is the natural \(n\)-dimensional representation) are complete intersections, and conjectured that they are in fact Gorenstein.
scientific article; zbMATH DE number 6336312
scientific article