# Vector invariant fields of finite classical groups

## TL;DR

This paper establishes the existence of a finite generating set of homogeneous invariants for the invariant fields of classical groups acting on multiple copies of a vector space over finite fields, extending to various group types and characteristics.

## Contribution

It proves the existence of explicit generators for the invariant fields of classical groups acting on multiple copies of a vector space over finite fields, in all characteristics.

## Key findings

- Existence of a finite set of homogeneous invariants generating the invariant field.
- Results apply to general linear, special linear, symplectic, unitary, and orthogonal groups.
- Theorems hold in any characteristic for the respective groups.

## Abstract

Let $W$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$ of any characteristic and $mW$ denote the direct sum of $m$ copies of $W$. Let $\mathbb{F}_q[mW]^{{\rm GL}(W)}$ and $\mathbb{F}_q(mW)^{{\rm GL}(W)}$ denote the vector invariant ring and vector invariant field respectively where ${\rm GL}(W)$ acts on $W$ in the standard way and acts on $mW$ diagonally. We prove that there exists a set of homogeneous invariant polynomials $\{f_{1},f_{2},\ldots,f_{mn}\}\subseteq \mathbb{F}_q[mW]^{{\rm GL}(W)}$ such that $\mathbb{F}_q(mW)^{{\rm GL}(W)}=\mathbb{F}_q(f_{1},f_{2},\ldots,f_{mn}).$ We also prove the same assertions for the special linear groups and the symplectic groups in any characteristic, and the unitary groups and the orthogonal groups in odd characteristic.