Vector invariant fields of finite classical groups

Abstract

Let $W$ be an $n$-dimensional vector space over a finite field ${\mathbb{F}}_q$ of any characteristic and $mW$ denote the direct sum of $m$ copies of $W$. Let ${\mathbb{F}}_q[mW{]}^{\mathrm{GL}(W)}$ and ${\mathbb{F}}_q(mW{)}^{\mathrm{GL}(W)}$ denote the vector invariant ring and vector invariant field respectively where $\mathrm{GL}(W)$ acts on $W$ in the standard way and acts on $mW$ diagonally. We prove that there exists a set of homogeneous invariant polynomials $\{f_1,f_2,\dots ,f_{mn}\}\subseteq {\mathbb{F}}_q[mW{]}^{\mathrm{GL}(W)}$ such that ${\mathbb{F}}_q(mW{)}^{\mathrm{GL}(W)}={\mathbb{F}}_q(f_1,f_2,\dots ,f_{mn}).$ We also prove the same assertions for the special linear groups and the symplectic groups in any characteristic, and the unitary groups and the orthogonal groups in odd characteristic.

15 pages; some errors have been corrected