\(SL_k\)-tilings are bi-infinite arrays \((a_{ij})_{i,j \in \mathbb{Z}}\) with values \(a_{ij} \in \mathbb{K}\) having the property that all \(k \times k\) adjacent minors are equal to one. This paper introduces and initiates the study of such objects. This notion is a natural generalization of \(SL_2\)-tilings defined by I. Assem, C. Reutenauer and D. Smith [Adv. Math. 225, No. 6, 3134–3165 (2010; Zbl 1275.13017)] in their study of friezes and their relation to cluster algebras and T-systems.
The \(SL_k\)-tilings of minimal rank \(k\) are of particular interest and are called tame. An explicit construction of all tame \(SL_k\)-tilings is given in terms of certain linearization coefficients. Several nice properties are shown, in particular the existence for every tame \(SL_k\)-tiling, of a dual \(SL_k\)-tiling obtained by computing the \((k-1)\times (k-1)\) minors. The authors also construct partial \(SL_k\)-tilings from admissible paths in \(\mathbb{Z} \times \mathbb{Z}\). Under certain conditions, these partial tilings may be extended in a unique way to complete tame \(SL_k\)-tilings, whose values exhibit Laurent phenomenon.
The main application given in the article concerns the frieze patterns of Coxeter, which are realized in terms of \(SL_2\)-tilings. New proofs of the main results of J. H. Conway and H. S. M. Coxeter [Math. Gaz. 57, 87–94 (1973; Zbl 0285.05028)] such as periodicity and glided-symmetry of frieze patterns are rederived in this context. Some properties of \(SL_k\)-tilings are also put in relation with T-systems appearing in mathematical physics.