Rhombus Filtrations and Rauzy Algebras

TL;DR

This paper develops techniques to analyze rhombus-based algebras from plane tilings, relating infinite and finite cases, with a focus on the self-similar Rauzy tiling to reveal algebraic properties.

Contribution

It introduces a filtration method using rhombus modules and connects infinite-dimensional algebras to finite ones, applied specifically to Rauzy tilings.

Findings

Identification of rhombus modules for algebra analysis

Relation between infinite and finite-dimensional algebras

Discovery of tree classes in the stable Auslander-Reiten quiver

Abstract

Peach introduced rhombal algebras associated to quivers given by tilings of the plane by rhombi. We develop general techniques to analyse rhombal algebras, including a filtration by what we call rhombus modules. We introduce a way to relate the infinite-dimensional rhombal algebra corresponding to a complete tiling of the plane to finite-dimensional algebras corresponding to finite portions of the tiling. Throughout, we apply our general techniques to the special case of the Rauzy tiling, which is built in stages reflecting an underlying self-similarity. Exploiting this self-similar structure allows us to uncover interesting features of the associated finite-dimensional algebras, including some of the tree classes in the stable Auslander-Reiten quiver.