On the combinatorics of rigid objects in 2-Calabi-Yau categories
Abstract
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 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.
Cite
@article{arxiv.0709.0882,
title = {On the combinatorics of rigid objects in 2-Calabi-Yau categories},
author = {Raika Dehy and Bernhard Keller},
journal= {arXiv preprint arXiv:0709.0882},
year = {2008}
}
Comments
11 pages, introduction expanded, references updated, to appear in IMRN