Coxeter elements and periodic Auslander–Reiten quiver

In this paper we show that for a simply-laced root system a choice of a Coxeter element C gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to C-orbits in R; moreover, it gives an identification of R with a certain subset of I×Z2h, where h is the Coxeter number. The set has a natural quiver structure; we call it the periodic Auslander–Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram I: roots are vertices of , and the root lattice and the inner product admit an explicit description in terms of . Finally, we relate this construction to the theory of quiver representations.