Titles and abstracts



Lidia Angeleri Huegel,   Localizing hearts of t-structures

The category of quasi-coherent sheaves over the projective line can be viewed as the heart of a t-structure induced by the ``largest'' infinite dimensional cotilting module over the Kronecker algebra A, and the Serre localizations of this heart correspond bijectively to the equivalence classes of infinite dimensional cotilting A-modules. A similar phenomenon occurs in more general contexts. We will discuss how to exploit this in connection with classification problems. The talk will be based on joint work with Javier Sanchez and on work in progress with Dirk Kussin.

Aslak Bakke Buan,   Torsion pairs and rigid objects in tubes

The talk is based on joint work with Karin Baur and Robert Marsh.
Tube categories are hereditary abelian category, arising as certain full subcategories of categories of finite dimensional modules over tame hereditary algebras. They are so-called because their Auslander-Reiten quivers have the shape of a tube. We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in an extension of the tube category containing the Prüfer and adic modules.
The annulus model for the tube category extends to this larger category, and we interpret torsion pairs, maximal rigid objects and the bijection between, in this model.

Mikhail Gorsky,   Distinguished bases for A_n root systems and parking functions

A well-known result of O. Lyashko and E. Looijenga states that the number of the distinguished bases of positive roots for A_n root system equals to (n+1)^{n-1}. I will give a combinatorial description of these bases by constructing an explicit bijection with the set of parking functions on n elements. I will describe the induced action of the braid group on parking functions and show natural links of these constructions to exceptional sequences in the category of representations of the Dynkin quiver A_n with standard orientation. For non-decreasing parking functions this braid group action is closely related to flips on Young diagrams which correspond to mutations in cluster algebras of type A_{n-1}. This is joint work with Evgeny Gorsky.

Sira Gratz,   Cluster Algebras of Infinite Rank. Slides of the talk

(see pdf-file here for an abstract using math symbols...)
The combinatorics of a cluster algebra of type Q, where Q is an orientation of the Dynkin diagram A_n, can be expressed via triangulations of the (n + 3)-gon. It follows that there is a cluster algebra structure of type Q on the homogeneous coordinate ring C[cone(Gr(2, n+3))] over the affine cone of the Grassmannian of planes in C^{n+3} via the Plücker embedding, as noted by Fomin and Zelevinsky. By allowing infinite countable clusters, this idea can be extended to the infinite case, motivated by results by Holm and Joergensen, who have analysed a category, whose cluster tilting subcategories correspond to triangulations of the infinty-gon. We thus get infinite cluster algebra structures on C[cone(Gr(2, +- infinity))], where Gr(2, +- infinity) is a direct limit of Grassmannians of planes in finite space. Moreover, the results of Grabowski and Launois on the quantum algebra structure on the quantum Grassmannian C_q[cone(Gr(2,n))] can be generalized to the infinite case, yielding infinite quantum cluster algebra structures on C_q [cone(Gr(2, +- infinite))].

Thorsten Holm,   Higher cluster tilting in Dynkin type A infinity

(Joint work with Peter Jorgensen)
For any integer d>0 there is an algebraic triangulated category C generated by a (d+1)-spherical object. These (d+1)-Calabi-Yau categories have been studied by various authors from different angles. Their Auslander-Reiten quivers consist of d components of type A infinity. In this talk we shall present a combinatorial classification of d-cluster tilting subcategories and weakly d-cluster tilting subcategories (lacking functorial finiteness) of these categories. We also describe the mutation behaviour of these subcategories which can be very different. On the one hand, in the d-cluster tilting subcategories each indecomposable object has exactly d mutations. On the other hand, for any l=0,...,d-1 there exist weakly d-cluster tilting subcategories having indecomposable objects with precisely l mutations. The combinatorics behind the classification of the (weakly) d-cluster tilting subcategories is that of (d+2)-angulations of the infinity-gon.

Bernhard Keller,   On cluster monomials

This is joint work with Giovanni Cerulli Irelli, Daniel Labardini-Fragoso and Pierre-Guy Plamondon. We show that for the cluster algebra associated with a (non valued) quiver, the cluster monomials are linearly independent. This confirms a conjecture by Fomin-Zelevinsky. We use categorification as developed in this generality by Pierre-Guy Plamondon.

Sefi Ladkani,   On mutation classes of quivers with constant number of arrows and derived equivalences

We characterize the quivers with the property that performing arbitrary sequences of Fomin-Zelevinsky mutations does not change their number of arrows. We also show that to each such quiver there is a naturally associated potential such that performing arbitrary sequences of QP mutations does not change the derived equivalence class of the corresponding Jacobian algebra. Thus, for these quivers (and potentials), mutation at ANY vertex behaves quite like the BGP reflection both combinatorially and algebraically. It turns out that these quivers arise from ideal triangulations of certain marked bordered surfaces. Most of the associated Jacobian algebras are finite-dimensional and gentle, but some of them are infinite-dimensional and locally gentle. The latter resemble the 3-Calabi-Yau algebras despite not being so.

Bernard Leclerc ,   Quantum cluster algebras and quantum coordinate rings

With every element w of the Weyl group of a symmetric Kac-Moody algebra, Lusztig and De Concini-Kac-Procesi have attached a quantum algebra A_q(n(w)). For example, if w is the longest element of the Weyl group of a simple Lie algebra g of type A,D,E, A_q(n(w)) is a quantum analogue of the coordinate ring of a maximal unipotent subgroup N of a complex algebraic group G with Lie algebra g. The talk will explain how A_q(n(w)) can be endowed with the structure of quantum cluster algebra, in the sense of Berenstein and Zelevinsky. This is a joint work with Christof Geiss and Jan Schröer.

Robert Marsh,   Categorification of a frieze pattern determinant

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

David Paukzstello,   Generalised Moore Spectra in a triangulated category

In this talk we construct a functor from a triangulated category which "approximates" each object of the triangulated category with an object from a module category. We show that this functor is well-behaved, and as an example of the theory, we recover Keller's canonical embedding of the module category of a path algebra of a quiver with no oriented cycles into its u-cluster category for u \geq 2.

Idun Reiten,   Pseudo-tilting theory

The talk is base on joint work with Osamu Iyama. We investigate a generalisation of tilting modules of projective dimension at most 1, which we call (support) pseudo-tilting modules. We show that the alsomst complete one have exactly 2 complements. The work was inspired by investigating connections between cluster categories and cluster-tilted algebras.

Raquel Simoes,   Hom-configurations and non-crossing partitions

Let Q be a Dynkin quiver and D^b(Q) the bounded derived category of the path algebra associated to Q. We will give a bijection between maximal Hom-free sets of indecomposable objects (the Hom-configurations in the title) in a certain orbit category of D^b(Q) and noncrossing partitions in the Weyl group associated to Q which are not contained in any proper standard parabolic subgroup.

Evgeny Smirnov,   Schubert polynomials, pipe dreams and related combinatorial objects
slides of the talk

Schubert polynomials were introduced by A.Lascoux and M.P.Schuetzenberger as a tool for studying the cohomology ring of a full flag variety. They naturally generalize well-known Schur polynomials. In 1996 S.Fomin and An.Kirillov provided a realization for Schubert polynomials using combinatorial objects usually referred to as pipe dreams, or rc-graphs.

It turns out that these pipe dreams have many interesting combinatorial properties which relate them, among others, to plane partitions (=three-dimensional Young diagrams) and Stasheff associahedra. Some of them were already observed in 1990s by Fomin and Kirillov, Billey et al.; some other properties are new. In my talk I will give an overview of these properties. Time permitting, I will also explain how pipe dreams are related to our work with V.Kiritchenko and V.Timorin on the realization of Schubert calculus on full flag varieties via Gelfand-Zetlin polytopes.

Hermund Torkildsen,   Geometric descriptions of m-cluster categories and coloured quiver mutation

Geometric descriptions of cluster categories have been studied by many authors (for example: Baur, Caldero, Chapoton, Lamberti, Marsh, Schiffler, Zhang). Baur and Marsh gave geometric descriptions of m-cluster categories of Dynkin type A and D. In this talk we will discuss the \tilde{A} case. In this setting we will also consider coloured quiver mutation as defined by Buan and Thomas. At the end we will briefly look at the \tilde{D} case.

Matthias Warkentin,   On mutation graphs of quivers

Let Q be an acyclic quiver and K an algebraically closed field. The exchange graph of tilting modules over KQ introduced by Riedtmann and Schofield has been studied extensively by Happel and Unger. After the introduction of cluster algebras and cluster categories it has been shown that this exchange graph can be seen as a part of the exchange graph of the cluster algebra given by Q, which is governed by the combinatorics of quiver mutations. We show (generalizing work of Assem, Blais, Bru ̈stle and Samson) that elementary considerations about quiver mutations yield new results about the corresponding exchange graphs. In particular we show that an exchange graph (of a cluster algebra) contains a cycle if and only if there are two vertices in Q that are not connected by more than one arrow. Furthermore we can classify all exchange graphs for quivers with three vertices answering a question by Happel in this case.