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Yann Palu

Publications and source records attributed to Yann Palu.

21 records · Page 2Linked to original sources
arXiv2010

A Caldero-Chapoton map for infinite clusters

We construct a Caldero-Chapoton map on a triangulated category with a cluster tilting subcategory which may have infinitely many indecomposable objects. The map is not necessarily defined on all objects of the triangulated category, but we show that it is a (weak) cluster map in the sense of Buan-Iyama-Reiten-Scott. As a corollary, it induces a surjection from the set of exceptional objects which can be reached from the cluster tilting subcategory to the set of cluster variables of an associated cluster algebra. Along the way, we study the interaction between Calabi-Yau reduction, cluster structures, and the Caldero-Chapoton map. We apply our results to the cluster category D of Dynkin type A infinity which has a rich supply of cluster tilting subcategories with infinitely many indecomposable objects. We show an example of a cluster map which cannot be extended to all of D. The case of D also permits us to illuminate results by Assem-Reutenauer-Smith on SL_2-tilings of the plane.

math.RT↗
arXiv2010

Cluster characters for triangulated 2-Calabi--Yau categories

Starting from an arbitrary cluster-tilting object $T$ in a 2-Calabi--Yau category over an algebraically closed field, as in the setting of Keller and Reiten, we define, for each object $L$, a fraction $X(T,L)$ using a formula proposed by Caldero and Keller. We show that the map taking $L$ to $X(T,L)$ is a cluster character, i.e. that it satisfies a certain multiplication formula. We deduce that it induces a bijection, in the finite and the acyclic case, between the indecomposable rigid objects of the cluster category and the cluster variables, which confirms a conjecture of Caldero and Keller.

math.RT↗