Let \(Q\) be a quiver without any loops or oriented cycles and let \(k\) be a field. Motivated by R. Marsh, M. Reineke and A. Zelevinsky [Trans. Am. Math. Soc. 355, No. 10, 4171–4186 (2003; Zbl 1042.52007)], the cluster category associated to the path algebra \(kQ\) was defined by P. Caldero, F. Chapoton and R. Schiffler [Trans. Am. Math. Soc. 358, No. 3, 1347–1364 (2006; Zbl 1137.16020)] (for the type \(A\) case) and A. B. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov [Adv. Math. 204, No. 2, 572–618 (2006; Zbl 1127.16011 )] (for the general case) in order to model the cluster algebras of Fomin and Zelevinsky. Cluster algebras were introduced in order to study the (dual) canonical/global crystal basis of a quantum group and total positivity in algebraic groups.
Motivated by the above articles, in this article it is shown that, if \(Q\) is of Dynkin type it is possible to give a direct definition of the cluster algebra in terms of the corresponding cluster category. In particular, a formula is given for the cluster variable corresponding to each (necessarily exceptional) indecomposable module over \(kQ\). The formula involves the Euler characteristics of generalised Grassmannians of quiver representations: varieties of subrepresentations of a representation \(M\) all having a fixed dimension vector.
It follows that all cluster variables can be described using the geometric representation theory of \(Q\) and the authors thus obtain the cluster algebra in terms of the module category of \(kQ\). See also P. Caldero and B. Keller [From triangulated categories to cluster algebras, preprint arXiv:math/RT/0506018v2 (2005), to appear in Invent. Math.], where the cluster algebra is expressed as a kind of Hall algebra of the corresponding cluster category.
At the end of the article, an intriguing connection between Coxeter-Conway friezes – see J. H. Conway and H. S. M. Coxeter [Math. Gaz. 57, 87–94 (1973; Zbl 0285.05028)] – and quiver representations and cluster algebras is described.