2015/05/22 by Lutz Hille, Hille, Lutz · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1505.06011
11 pages
arxiv created 2015/05/22 · openalex publication_date 2015/05/22 · arxiv updated 2015/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well known that the number of tilting modules over a path algebra of type An coincides with the Catalan number C(n). Moreover, the number of support tilting modules of type An is the Catalan number C(n+1). We show that the convex hull of all roots of a root system of type An is a polytope with integral volume (n + 1)C(n+1). 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.