On invariant fields of vectors and covectors

Abstract: Let ${\mathbb{F}{q}}$ be the finite field of order $q$. Let $G$ be one of the three groups ${\rm GL}(n, \mathbb{F}_q)$, ${\rm SL}(n, \mathbb{F}_q)$ or ${\rm U}(n, \mathbb{F}_q)$ and let $W$ be the standard $n$-dimensional representation of $G$. For non-negative integers $m$ and $d$ we let $mW\oplus d W^*$ denote the representation of $G$ given by the direct sum of $m$ vectors and $d$ covectors. We exhibit a minimal set of homogenous invariant polynomials ${\ell_1,\ell{2},\dots,\ell_{(m+d)n}}\subseteq \mathbb{F}q[mW\oplus d W^*]^G$ such that $\mathbb{F}_q(mW\oplus d W^*)^G=\mathbb{F}_q(\ell_1,\ell_2,\dots,\ell{(m+d)n})$ for all cases except when $md=0$ and $G={\rm GL}(n, \mathbb{F}_q)$ or ${\rm SL}(n, \mathbb{F}_q)$.