Rational invariants of certain classical groups over finite fields

Abstract

Let $\mathbb{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 $\mathbb{F}_q $ and $\mathbb{F}_q $ be the rational function field over $\mathbb{F}_q $. We seek to understand the structure of the rational invariant subfield $\mathbb{F}_q $. In this paper, we prove that $\mathbb{F}_q $ 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.

Keywords

Rational invariant / Dickson invariant / rational function field / symplectic group

References

[1] Carlisle D., Kropholler P. H. Rational invariants of certain orthogonal and unitary groups. Bull London Math Soc, 1992, 24: 57-60.

[2] Chu H. Supplementary note on ‘rational invariants of certain orthogonal and unitary groups’. Bull London Math Soc, 1997, 29: 37-42.

[3] Dickson L. E. A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans Amer Math Soc, 1911, 12: 75-98.

[4] Kemper G. Calculating invariant rings of finite groups over arbitrary fields. J Symbo Computation, 1996, 21: 351-366.

[5] Rajaei S. M. Rational invariants of certain orthogonal groups over finite of characteristic two. Comm Algebra, 2000, 28: 2367-2393.

[6] Tang Z., Wan Z. A matrix approach to the rational invariants of certain classical groups over finite fields of characteristic two. Finite Fields and Their Applications, 2006, 12: 186-210.

[7] Wan Z. Geometry of Classical Groups over Finite Fields, 2002 2nd ed. Beijing: Science Press.