10:00am - 12:00pmCluster algebras and positivity
Chair(s): Lisa Lamberti (ETHZ, Switzerland), Khrystyna Serhiyenko (University of California, Berkeley, USA / University of Kentucky, Lexington), Lauren Williams (Harvard, USA)
Cluster algebras are commutative rings whose generators and relations can be defined in a remarkably succinct recursive fashion. Algebras of this kind, introduced by Fomin and Zelevinsky in 2000, are equipped with a powerful combinatorial structure frequently appearing in many mathematical contexts such as Lie theory, triangulations of surfaces, Teichmueller theory and beyond. Coordinate rings of Grassmannians and related invariant rings are well-studied examples of algebras of this type. One important aspect arising from the intrinsic combinatorial structure of cluster algebras is that it uncovers systematic, intriguing and complex positivity properties in these families of rings. For instance, it is expected that for each cluster algebra there is a distinguished basis, such that all elements can be expressed as a "positive" linear combination of basis vectors. Seemingly elementary claims of this type, so far proved only in certain cases, have triggered important developments in research areas at the intersection of geometry, algebra and combinatorics.
In this session, we glimpse at recent developments in this field and discuss open questions.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Combinatorics of cluster structures in Schubert varieties
Khrystyna Serhiyenko1, Melissa Sherman-Bennett2, Lauren Williams3
1University of California, Berkeley, USA / University of Kentucky, Lexington, 2University of California, Berkeley, USA, 3Harvard, USA
The (affine cone over the) Grassmannian is a prototypical example of a variety with cluster structure. Scott (2006) gave a combinatorial description of this cluster algebra in terms of Postnikov's plabic graphs. It has been conjectured essentially since Scott's result that Schubert varieties also have a cluster structure with a description in terms of plabic graphs. I will discuss recent work with K. Serhiyenko and L. Williams proving this conjecture. The proof uses a result of Leclerc, who shows that many Richardson varieties in the full flag variety have cluster structure using cluster-category methods, and a construction of Karpman to build plabic graphs for each Schubert variety.
Cluster tilting modules for mesh algebras
Karin Erdmann1, Sira Gratz2, Lisa Lamberti3
1University of Oxford, UK, 2University of Glasgow, UK, 3ETHZ, Switzerland
Mesh algebras are a class of finite-dimensional algebras which generalize naturally preprojective algebras. In this talk, I describe cluster tilting modules for mesh algebras of Dynkin type and discuss possible relations to skew-symmetrizable cluster algebras structures in certain coordinate rings. This is joint work with K. Erdmann and S. Gratz.
Strings, snake graphs and the cluster expansion formulas
Ilke Canakci1, Vincent Pilaud2, Nathan Reading3, Sibylle Schroll4
1Newcastle University, 2École polytechnique, 3NCSU Campus, 4University of Leicester, UK
Snake graphs arise from cluster algebras associated to triangulations of marked oriented surfaces in the work of Musiker, Schiffler and Williams in the context of Laurent expansion formulas. In this talk we will show a correspondence between snake graphs and combinatorial objects called strings. String combinatorics has first arising in the context of the classification of indecomposable modules in a large class of tame algebras, the so-called special biserial algebras. We will show how this new interpretation of snake graphs in terms of strings leads to an alternative cluster expansion formula for cluster algebras arising from triangulations of surfaces. This is joint work with Ilke Canakci as well as joint work in progress with Nathan Reading and Vincent Pilaud.
Friezes and Grassmannian cluster structures
Karin Baur1, Eleonore Faber2, Sira Gratz3, Khrystyna Serhiyenko4, Gordana Todorov5
1Universität Graz / University of Leeds, Austria / UK, 2University of Leeds, UK, 3University of Glasgow, UK, 4University of California, Berkeley, USA / University of Kentucky, Lexington, 5Northeastern University, Boston, USA
In this talk, I will show how to obtain SL_k-friezes using Plücker coordinates by taking certain subcategories of the Grassmannian cluster categories. These are cluster structures associated to the Grassmannians of k-spaces in n-space. Many of these friezes arise from specialising a cluster to 1. We use Iyama-Yoshino reduction to reduce the rank of such friezes.
This is joint work with E. Faber, S. Gratz, G. Todorov, K. Serhyenko. https://arxiv.org/abs/1810.10562