2007/12/10 by Yu Zhou, Bin Zhu, Zhou, Yu +1 · 1 citation
Mathematics · #05A15 #16G20 #16G70 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT #msc:05A15 #msc:16G20 #msc:16G70
paper · pdf · doi:10.48550/arxiv.0712.1381
correted many typos according to the referee's comments, final version to appear in J. Algebra
arxiv created 2009/02/14 · arxiv updated 2009/12/01
We study the cluster combinatorics of d-cluster tilting objects in d-cluster categories. By using mutations of maximal rigid objects in d-cluster categories which are defined similarly for d-cluster tilting objects, we prove the equivalences between d-cluster tilting objects, maximal rigid objects and complete rigid objects. Using the chain of d+1 triangles of d-cluster tilting objects in [IY], we prove that any almost complete d-cluster tilting object has exactly d+1 complements, compute the extension groups between these complements, and study the middle terms of these d+1 triangles. All results are the extensions of corresponding results on cluster tilting objects in cluster categories established in [BMRRT] to d-cluster categories. They are applied to the Fomin-Reading's generalized cluster complexes of finite root systems defined and studied in [FR2] [Th] [BaM1-2], and to that of infinite root systems [Zh3].