Cluster algebras and representation theory

University of Leeds, Thursday 17 June, 2010

Here is some information on getting to the School of Mathematics at Leeds.
The meeting is supported by the Engineering and Physical Sciences Research Council, and is organised by Yann Palu and Robert Marsh.



If you are planning to come, please send an email to Yann Palu.
If you need accommodation, the Cliff Lawn Hotel is recommended. See also the university list of accommodation.


Peter Jorgensen, A Caldero-Chapoton map for infinite clusters.
The Caldero-Chapoton map is a major mathematical achievement of the last decade. It formalises the connection between the cluster algebras of Fomin-Zelevinsky and the cluster categories of Buan-Marsh-Reineke-Reiten-Todorov by mapping objects of cluster categories to cluster variables in cluster algebras.

We will show a version of the Caldero-Chapoton map which works in the case of infinite clusters. The map is not necessarily defined on all objects of the category; this will be exemplified by the cluster category of Dynkin type A infinity.

Yann Palu, Coloured quivers of partial triangulations (and rigid objects in cluster categories).
In this talk I will mostly focus on examples.

In the theory of categorification of cluster algebras, cluster-tilting objects and maximal rigid objects play a prominent role. The need to consider (non-maximal) rigid objects recently appeared in the work of Plamondon. Independently, Buan and Marsh have defined and studied a mutation for arbitrary rigid objects in 2 Calabi-Yau categories.

In a joint work in progress with Robert Marsh, we consider partial triangulations of unpunctured compact Riemann surfaces with boundaries and marked points. We define a mutation for partial triangulations which generalizes the flip of triangulations, and, following Buan and Thomas, a coloured quiver. This mutation corresponds to the mutation of rigid objects in the cluster category (Amiot; Labardini-Fragoso; Keller-Yang) associated with the surface. Moreover, the coloured quiver of a partial triangulation coincides with a coloured quiver we define using the corresponding rigid object.

Pierre-Guy Plamondon, Hom-infinite cluster categories.
One of several approaches to the theory of cluster algebras is that of (additive) categorification. C. Amiot's generalized cluster category, defined for any quiver with potential, together with Y. Palu's cluster character, is known to categorify the associated cluster algebra, provided that the category is Hom-finite. In this talk, we will study the cluster category in the case where it is Hom-infinite. We will define a (non-triangulated) subcategory of the cluster category which is "almost 2-Calabi-Yau" and stable under mutation, and we will define a cluster character on it. This construction will allow us to recover the associated cluster algebra.

Robert Marsh, Categorifying the determinant of a frieze pattern of integers.
Joint work with Karin Baur (ETH, Zürich). Broline, Crowe and Isaacs have computed the determinant of a matrix associated to a Coxeter-Conway frieze pattern of integers. We generalise this to the corresponding frieze pattern of cluster variables arising from a Fomin-Zelevinsky cluster algebra of type A and give a representation-theoretic interpretation of this result in terms of configurations of indecomposable objects in the root category of a type A quiver.

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Page updated by Robert Marsh. Last modified 11 June 2010.