Rational invariants of certain classical groups over finite fields

Abstract

Let F_q be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over F_q and F_q(x₁, ..., x_n) be the rational function field over F_q. We seek to understand the structure of the rational invariant subfield F_q(x₁, ..., x_n)^G. In this paper, we prove that F_q(x₁, ..., x_n)^G is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.

References

1. Carlisle D, Kropholler P H . Rational invariants of certainorthogonal and unitary groups. Bull LondonMath Soc, 1992, 24: 57–60. doi:10.1112/blms/24.1.57
2. Chu H . Supplementarynote on ‘rational invariants of certain orthogonal and unitarygroups’. Bull London Math Soc, 1997, 29: 37–42. doi:10.1112/S0024609396001580
3. Dickson L E . A fundamental system of invariants of the general modular lineargroup with a solution of the form problem. Trans Amer Math Soc, 1911, 12: 75–98. doi:10.2307/1988736
4. Kemper G . Calculatinginvariant rings of finite groups over arbitrary fields. J Symbo Computation, 1996, 21: 351–366. doi:10.1006/jsco.1996.0017
5. Rajaei S M . Rational invariants of certain orthogonal groups over finite of characteristictwo. Comm Algebra, 2000, 28: 2367–2393. doi:10.1080/00927870008826966
6. Tang Zhongming, Wan Zhexian . A matrix approach to therational invariants of certain classical groups over finite fieldsof characteristic two. Finite Fields andTheir Applications, 2006, 12: 186–210. doi:10.1016/j.ffa.2005.04.005
7. Wan Zhexian . Geometry of Classical Groups over Finite Fields. 2nd ed. Beijing: Science Press, 2002