On invariant fields of vectors and covectors

TL;DR

This paper identifies minimal generating sets of invariant polynomials for vector and covector representations of certain finite groups over finite fields, expanding understanding of invariant theory in modular settings.

Contribution

It provides explicit minimal generating sets of invariant polynomials for the actions of GL, SL, and U groups on vector and covector representations over finite fields.

Findings

Explicit minimal generating sets for invariants are constructed.

Complete description of invariant fields for most cases, except when $md=0$ for GL and SL.

Advances the understanding of invariant theory over finite fields in modular cases.

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 ${\rm SL}(n, \mathbb{F}_q)$.