On the Combinatorics of Rigid Objects in 2-Calabi-Yau Categories
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Publication:3515968
DOI10.1093/IMRN/RNN029zbMATH Open1144.18009arXiv0709.0882OpenAlexW1980276774MaRDI QIDQ3515968
Author name not available (Why is that?)
Publication date: 30 July 2008
Published in: (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/0709.0882
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