On the combinatorics of rigid objects in 2-Calabi-Yau categories
Dehy, Raika ; Keller, Bernhard
arXiv, 0709.0882 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2007-09-06
Classification:  Mathematics - Representation Theory,  Mathematics - Rings and Algebras,  18E30 
@article{0709.0882,
     author = {Dehy, Raika and Keller, Bernhard},
     title = {On the combinatorics of rigid objects in 2-Calabi-Yau categories},
     journal = {arXiv},
     volume = {2007},
     number = {0},
     year = {2007},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0709.0882}
}

Dehy, Raika; Keller, Bernhard. On the combinatorics of rigid objects in 2-Calabi-Yau categories. arXiv, Tome 2007 (2007) no. 0, . http://gdmltest.u-ga.fr/item/0709.0882/