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Abstract: Given a triangulated 2-Calabi-Yau category C and a cluster-tiltingsubcategory T, the index of an object X of C is a certain element of theGrothendieck group of the additive category T. In this note, we show that arigid object of C is determined by its index, that the indices of theindecomposables of a cluster-tilting subcategory T- form a basis of theGrothendieck group of T and that, if T and T- are related by a mutation, thenthe indices with respect to T and T- are related by a certain piecewise lineartransformation introduced by Fomin and Zelevinsky in their study of clusteralgebras with coefficients. This allows us to give a combinatorial constructionof the indices of all rigid objects reachable from the given cluster-tiltingsubcategory T. Conjecturally, these indices coincide with Fomin-Zelevinsky-sg-vectors.



Autor: Raika Dehy, Bernhard Keller

Fuente: https://arxiv.org/



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