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 $\mathrm{GL}(n,{\mathbb{F}}_q)$, $\mathrm{SL}(n,{\mathbb{F}}_q)$ or $\mathrm{U}(n,{\mathbb{F}}_q)$ and let $W$ be the standard $n$-dimensional representation of $G$. For non-negative integers $m$ and $d$ we let $mW\oplus d{W}^{\ast }$ 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 $\{{\ell }_1,{\ell }_2,\dots ,{\ell }_{(m+d)n}\}\subseteq {\mathbb{F}}_q[mW\oplus d{W}^{\ast }{]}^G$ such that ${\mathbb{F}}_q(mW\oplus d{W}^{\ast }{)}^G={\mathbb{F}}_q({\ell }_1,{\ell }_2,\dots ,{\ell }_{(m+d)n})$ for all cases except when $md=0$ and $G=\mathrm{GL}(n,{\mathbb{F}}_q)$ or $\mathrm{SL}(n,{\mathbb{F}}_q)$.

Added a reference to the "No-name Theorem" and modified the introduction to account for this addition