Periodicity in bilinear lattices and the Coxeter formalism

We introduce and study in detail so-called circulant (Coxeter-periodic) elements and circulant families in a bilinear lattice K as well as their dual versions, called anti-circulant. We show that they form a natural environment for a systematic explanation of certain cyclotomic factors of the Coxeter polynomial χK of K and in consequence, of Coxeter polynomials of algebras of finite global dimension. We discuss the properties of quadratic forms induced by circulant and anti-circulant families. Moreover, we interpret the results in the language of representation theory of algebras and point out applications (facts concerning tubular families in Auslander-Reiten quivers and quadratic forms of algebras). Abstract considerations in bilinear lattices are illustrated with a collection of non-trivial examples arising from module and derived categories. The results show that techniques of linear algebra and number theory provide efficient tools for explaining various representation theoretic facts.