Frises

Each acyclic graph, and more generally, each acyclic orientation of the graph associated to a Cartan matrix, allows to define a so-called frise; this is a collection of sequences over the positive natural numbers, one for each vertex of the graph. We prove that if these sequences satisfy a linear recurrence, then the Cartan matrix is of Dynkin type (if the sequences are bounded) or of Euclidean type (if the sequences are unbounded). We prove the converse in all cases, except for the exceptional Euclidean Cartan matrices; we show even that the sequences are rational over the positive natural numbers. We generalize these results by considering frises with variables; as a byproduct we obtain, for the Dynkin and Euclidean type A cases, explicit formulas for the cluster variables, over the semiring of Laurent polynomials over the positive natural numbers generated by the initial variables (which explains simultaneously positivity and the Laurent phenomenon). The general tool are the so-called SL_2-tilings of the plane; these are fillings of the whole discrete plane by elements of a ring, in such a way that each 2 by 2 connected submatrix is of determinant 1.