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arXiv · 0709.0882

On the combinatorics of rigid objects in 2-Calabi-Yau categories

Raika Dehy·Bernhard Keller
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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.

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Raika Dehy, Bernhard Keller. 2008-04-14. On the combinatorics of rigid objects in 2-Calabi-Yau categories. https://arxiv.org/abs/0709.0882

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