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 ${ \rm SL}(n, \mathbb{ F}_q)$.
Categories: math.AC