Invariants of the dihedral group $D_{2p}$ in characteristic two

TL;DR

This paper investigates the invariants of dihedral groups in characteristic two, providing bounds on generating degrees, identifying minimal separating sets, and explicitly describing these sets for prime p.

Contribution

It offers new bounds on generating degrees independent of p for large representations and characterizes minimal separating sets for dihedral groups in characteristic two.

Findings

Upper bound for degrees of generators independent of p

p+1 is the minimal degree for separating sets

Explicit description of separating sets when p is prime

Abstract

We consider finite dimensional representations of the dihedral group $D_{2p}$ over an algebraically closed field of characteristic two where $p$ is an odd integer and study the degrees of generating and separating polynomials in the corresponding ring of invariants. We give an upper bound for the degrees of the polynomials in a minimal generating set that does not depend on $p$ when the dimension of the representation is sufficiently large. We also show that $p+1$ is the minimal number such that the invariants up to that degree always form a separating set. As well, we give an explicit description of a separating set when $p$ is prime.