Tilting Modules over the Path Algebra of Type \Aa, Polytopes, and Catalan Numbers

Abstract

It is well known that the number of tilting modules over a path algebra of type An coincides with the Catalan number Cn. Moreover, the number of support tilting modules of type An is the Catalan number C n+1 . We show that the convex hull C(An) of all roots of a root system of type An is a polytope with integral volume (n + 1)C n+1 = 2n n . Moreover, we associate to the set of tilting modules and to the set of support tilting modules certain polytopes and show that their volumes coincide with the number of those modules, respectively. Finally, we show that these polytopes can be defined just using the root system and relate their volumes so that we can derive the above results in a new way.