$SL_k$-tilings of the plane

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.