# 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.

#### Schedule

- 10:45-11.15 :
Tea/coffee, Common Room (School of Mathematics Level 9).
- 11.15-12.45
**
Peter Jørgensen (Newcastle),
A Caldero-Chapoton map for infinite clusters.
**

Roger Stevens Lecture Theatre 11.
- 12.45-2.00 : Lunch.
- 2.00-3:00,
**
Robert Marsh (Leeds),
Categorifying the determinant of a frieze pattern of integers.
**

MALL 2, School of Mathematics Level 8.
- 3.15-4.15,
**
Yann Palu (Leeds),
Coloured quivers of partial triangulations
(and rigid objects in cluster categories).
**.

MALL 2, School of Mathematics Level 8.
- 4.15-4.45 : Tea/coffee, Common Room (School of Mathematics Level 9).
- 4.45-6.15,
**
Pierre-Guy Plamondon
(Institut de Mathématiques de Jussieu),
Hom-infinite cluster categories.
**

MALL 2, School of Mathematics Level 8.

#### Registration

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.

## Abstracts

**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.

[ University of Leeds ]
[ School of Mathematics ]
[ Department of Pure Maths ]

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