Seminars in Cluster Algebras

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Martin Frankland: Toda brackets in n-angulated categories

17th October, 2024

Geiss, Keller, and Oppermann introduced n-angulated categories to capture the structure found in certain cluster tilting subcategories in quiver representation theory. Jasso and Muro investigated Toda brackets and Massey products in such cluster tilting subcategories by using the ambient triangulated category. In joint work with Sebastian Martensen and Marius Thaule, we introduce Toda brackets in n-angulated categories, generalizing Toda brackets in triangulated categories (the case n = 3). We will look at different constructions of the brackets, their properties, some examples, and some applications.

Yann Palu: 0-Auslander extriangulated categories

13th October, 2023

Categorification of cluster algebras has instilled the idea of mutation in representation theory. Nice theories of mutation, for some forms of rigid objects, have thus been developed in various settings. In a collaboration with Mikhail Gorsky and Hiroyuki Nakaoka, we axiomatized the similarities between most of those settings under the name of 0-Auslander extriangulated categories. The prototypical example of a 0-Auslander extriangulated category is the category of two-term complexes of projectives over a finite-dimensional algebra. In this talk, we will give several examples of 0-Auslander categories, and explain how they relate to two-term complexes.

Khrystyna Serhiyenko: Leclerc’s conjecture on a cluster structure for type A Richardson varieties

1st May, 2023

Coordinate rings of many varieties naturally occurring in representation theory are known to admit a cluster algebra structure. Leclerc constructed a conjectural cluster structure on Richardson varieties using categorification in terms of module categories of the preprojective algebras. We show that in type A, his conjectural cluster structure is in fact a cluster structure. We do this by comparing Leclerc’s construction with another cluster structure due to Ingermanson, which uses the combinatorics of wiring diagrams and the Deodhar stratification.

Nicholas Williams: A structural view of maximal green sequences

27th April, 2023

We initiate a new approach to maximal green sequences, whereby they are considered under an equivalence relation. Doing this reveals extra structure on the set of maximal green sequences of an algebra, namely hidden partial orders. We prove that there are several different ways of defining the equivalence relation. Similarly, there are several ways of defining a partial order on the equivalence classes, but as yet these partial orders are only conjecturally equivalent. We prove this conjecture in the case of Nakayama algebras.

Jonah Berggren: Boundary algebras of positroids

27th April, 2023

The boundary algebra of a dimer model derived from a Postnikov diagram may be used to obtain an additive categorification of the cluster algebra of the associated positroid variety. The boundary algebra has an explicit description in the case of Grassmannians but not for more general positroids. It is known that every Postnikov diagram is move equivalent to one which comes from a Le-diagram and has an isomorphic boundary algebra. We use the perfect matching structure of a dimer model derived from a Le-diagram to provide an algorithm for computing the Gabriel quiver and relations of its boundary algebra.

Melissa Sherman-Bennett: Cluster structure on braid varieties

3rd April, 2023

Braid varieties for SLn are smooth affine varieties associated to any positive braid. Their cohomology contains information about the Khovanov-Rozansky homology of a related link. One can analogously define braid varieties for any simple algebraic group. Special cases of braid varieties include Richardson varieties, double Bruhat cells, and double Bott-Samelson cells. Cluster algebras, introduced by Fomin and Zelevinsky, are a class of commutative rings which are completely determined by a combinatorial input called a seed. I'll discuss joint work with P. Galashin, T. Lam and D. Speyer in which we show the coordinate rings of braid varieties are cluster algebras. In the SLn case, seeds for these cluster algebras come from '3D plabic graphs', which are bicolored graphs embedded in a 3-dimensional ball and generalize Postnikov's plabic graphs for positroid varieties.

Francesca Fedele: Ext-projectives in subcategories of triangulated categories

21st February, 2023

Let T be a suitable triangulated category and C a full subcategory of T closed under summands and extensions. An indecomposable object c in C is called Ext-projective if Ext1(c,C)=0. Such an object cannot appear as the endterm of an Auslander-Reiten triangle in C. However, if there exists a minimal right almost split morphism b→c in C, then the triangle x→b→c→ extending it is a so called left-weak Auslander-Reiten triangle in C. We show how in some cases removing the indecomposable c from the subcategory C and replacing it with the indecomposable x gives a new extension closed subcategory C' of T and see how this operation is related to Iyama-Yoshino mutation of C with respect to a rigid subcategory. Time permitting, we will see the application of the result to cluster categories of type A.

Bernhard Keller: On Amiot’s conjecture

17th January, 2023

In 2010, Claire Amiot conjectured that algebraic 2-Calabi-Yau categories with a cluster-tilting object must come from quivers with potential. This would extend a structure theorem obtained with Idun Reiten in the case where the endomorphism algebra of the cluster-tilting object is hereditary. Many other classes of examples are also known. We will report on recent progress in the general case obtained in joint work with Junyang Liu and based on Van den Bergh's structure theorem for complete Calabi-Yau algebras.

Roger Casals: A Microlocal Invitation to Lagrangian Fillings

11th November, 2022

We present recent developments in symplectic geometry and explain how they motivated new results in the study of cluster algebras. First, we introduce a geometric problem: the study of Lagrangian surfaces in the standard symplectic 4-ball bounding Legendrian knots in the standard contact 3-sphere. Thanks to results from the microlocal theory of sheaves, which we will survey, we then show that this geometric problem gives rise to an interesting moduli space. In fact, we establish a bridge translating geometric operations, such as Lagrangian disk surgeries, into algebraic properties of this moduli space, such as the existence of cluster algebra structures. The talk is intended for a broad symplectic audience and all key ideas will be introduced and motivated.