Polynomial invariants of finite unitary groups
The present paper deals with the problem of finding the invariants of classical groups over finite fields. The earlier result on this was due to \textit{L. E. Dickson} [American M. S. Trans. 12, 75--98 (1911; JFM 42.0136.01)], which treated the cases of general linear and special linear groups. Another important paper is that of \textit{D. Carlisle} and \textit{P. H. Kropholler} [Bull. Lond. Math. Soc. 24, 57--60 (1992; Zbl 0778.12006)], which explicit generators for the rational invariants of orthogonal and unitary groups, and also found explicit generators and relations for the rings of polynomial invariants of symplectic groups. One should mention that their results showed that these invariant rings are all complete intersections. As for the invariant rings of orthogonal and unitary groups there are several papers dealing with low-dimensional cases and a recent paper [\textit{P. H. Kropholler, S. Mohseni Rajaei} and \textit{J. Segal}, Glasg. Math. J. 47, No. 1, 7--54 (2005; Zbl 1103.13007)] treats the orthogonal groups over \(\mathbb{F}_{2}\), the general case remains open. Let \(U_{n}(\mathbb{F}_{q^{2}})\) be the \(n\)-dimensional unitary group over the finite field \(\mathbb{F}_{q^{2}}\). The goal of this paper is to contribute in solving the above problem, by finding generators and relations for the ring of invariants of \(U_{n}(\mathbb{F}_{q^{2}})\). There is also prove that this is a complete intersection.
scientific article; zbMATH DE number 5078143