| Résume | Cluster algebras were recently defined by S. Fomin and A. Zelevinsky in an attempt to describe the Lusztig-Kashiwara canonical basis of a quantum group, and to develop an algebraic framework for total positivity in semisimple algebraic groups. In this work they associate to each finite root system a complete simplicial fan which, according to work by F. Chapoton, S. Fomin and A. Zelevinsky, is the normal fan of a polytope which can be regarded as a generalised version of the associahedron, or Stasheff polytope (in type $A$, it is this polytope). We provide a quiver-theoretic interpretation of these fans (and, in particular, cluster combinatorics); their main properties then become easy consequences of the known facts about tilting modules over the path algebras of Dynkin quivers due to K. Bongartz, D. Happel and C. M. Ringel. |
