Separating invariants over finite fields

TL;DR

This paper determines the minimal number of separating invariants needed for matrix group invariants over finite fields, providing degree bounds and specific examples for symmetric groups over e2b9b2.

Contribution

It establishes explicit bounds on the degrees of separating invariants for finite field matrix groups, improving known results in the non-modular case.

Findings

Minimal number of separating invariants determined

Degree bounds for invariants established

Examples for symmetric group representations over e2b9b2 provided

Abstract

We determine the minimal number of separating invariants for the invariant ring of a matrix group $G < \mathrm{GL}_n(\mathbb{F}_q)$ over the finite field $\mathbb{F}_q$. We show that this minimal number can be obtained with invariants of degree at most $|G|n(q-1)$. In the non-modular case this construction can be improved to give invariants of degree at most $n(q-1)$. As examples we study separating invariants over the field $\mathbb{F}_2$ for two important representations of the symmetric group