On modular vector invariant fields

Abstract

Let ${\mathbb{F}}_q$ be a finite field of any characteristic and $\mathrm{GL}(\mathrm{n},{\mathbb{F}}_{\mathrm{q}})$ the general linear group over ${\mathbb{F}}_q$. Suppose $W$ denotes the standard representation of $\mathrm{GL}(\mathrm{n},{\mathbb{F}}_{\mathrm{q}})$ and $\mathrm{GL}(\mathrm{n},{\mathbb{F}}_{\mathrm{q}})$ acts diagonally on the direct sum of $W$ and its dual space $W^{\ast }$. Let $G$ be any subgroup of $\mathrm{GL}(\mathrm{n},{\mathbb{F}}_{\mathrm{q}})$. 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 ${\mathbb{F}}_q[W\oplus W^{\ast }{]}^G$ such that the invariant field ${\mathbb{F}}_q(W\oplus W^{\ast }{)}^G$ is generated by $\{f_1,f_2,\dots ,f_k,{\ell }_1,{\ell }_2,\dots ,{\ell }_n\}$ over ${\mathbb{F}}_q$.

arXiv comment5 pages