$SL_k$-Tiling of the Plane

TL;DR

This paper investigates bi-infinite arrays with specific minor conditions, revealing their rank properties, explicit constructions, and connections to Coxeter frieze patterns and T-systems in statistical physics.

Contribution

It characterizes arrays with all adjacent k×k minors equal to one and all (k-1)×(k-1) minors nonzero, providing explicit constructions and new insights into Coxeter frieze patterns and T-systems.

Findings

Arrays with specified minor conditions have rank k.

Explicit construction methods for these arrays are provided.

Connections established between array properties, Coxeter frieze patterns, and T-systems.

Abstract

We study properties of (bi-infinite) arrays having all adjacent $k\times k$ adjacent minors equal to one. If we further add the condition that all adjacent $(k-1)\times (k-1)$ minors be nonzero, then these arrays are necessarily of rank $k$. It follows that we can explicit construct all of them. Several nice properties are made apparent. In particular, we revisit, with this perspective, the notion of frieze patterns of Coxeter. This shed new light on their properties. A connexion is also established with the notion of $T$-systems of Statistical Physics.