Actions of frieze groups on inverse limits of polynomial rings

TL;DR

This paper explores how frieze groups act on rings of infinite polynomial combinations, describing invariant subrings and module structures, extending ideas from symmetric group actions.

Contribution

It introduces the actions of all seven frieze groups on infinite polynomial rings and characterizes their invariant subrings and module structures.

Findings

Explicit descriptions of invariant subrings for each frieze group

Analysis of the rings as modules over the frieze groups

Extension of symmetric group action concepts to frieze groups

Abstract

In the spirit of the action of the symmetric group on the ring of polynomials in $n$ variables, we consider the actions of the seven frieze groups on rings of formal infinite linear combinations of monomials of restricted degree. For each group we describe the respective subring of invariants. We discuss also the structure of those rings as modules over each frieze group.