On the Combinatorics of Rigid Objects in 2-Calabi-Yau Categories

R. Dehy, B. Keller
2010 International mathematics research notices  
Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T , the index of an object X of C is a certain element of the Grothendieck group of the additive category T . In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a cluster-tilting subcategory T form a basis of the Grothendieck group of T and that, if T and T are related by a mutation, then the indices with respect to T and T are related by a certain
more » ... ted by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given cluster-tilting subcategory T . Conjecturally, these indices coincide with Fomin-Zelevinsky's g-vectors.
doi:10.1093/imrn/rnn029 fatcat:e53acqfhpfgdhdb657xcdl7wni