On modular vector invariant fields

Abstract

Let $\mathbb{F}_q$ be a finite field of any characteristic and $\rm GL(n,\mathbb{F}_q)$ the general linear group over $\mathbb{F}_q$. Suppose $W$ denotes the standard representation of $\rm GL(n,\mathbb{F}_q)$ and $\rm GL(n,\mathbb{F}_q)$ acts diagonally on the direct sum of $W$ and its dual space $W^*$. Let $G$ be any subgroup of $\rm GL(n,\mathbb{F}_q)$. Suppose the invariant field $\mathbb{F}_q(W)^G=\mathbb{F}_q(f_1,f_2,\dots,f_k)$, where $f_1,f_2,\dots,f_k\in\mathbb{F}_q[W]^G$ are homogeneous invariant polynomials. We prove that there exist homogeneous polynomials $\ell_1,\ell_2,\dots,\ell_{n}$ in the invariant ring $\mathbb{F}_q[W\oplus W^*]^{G}$ such that the invariant field $\mathbb{F}_q(W\oplus W^*)^{G}$ is generated by $\{f_1,f_2,\dots,f_k,\ell_1,\ell_2,\dots,\ell_{n}\}$ over $\mathbb{F}_q$.