In cluster theory, establishing explicit formulae for cluster variables is an important problem. In the setting of surface cluster algebras this may be given combinatorially via snake graphs and homologically by the CC-map. Recently a combinatorial formula was introduced by Musiker--Ovenhouse--Zhang in an attempt to introduce super cluster algebras of type A (which computes super lambda-lengths in Penner-Zeitlin's super-Teichmüller spaces). This formula is given in terms of double dimer covers on snake graphs. Motivated by this construction, we propose a representation theoretic interpretation of super lambda-lengths and introduce a super-CC formula which recovers the combinatorial model. This is joint work in progress with Fedele, Garcia Elsener and Serhiyenko.
Stability conditions are commonly used in the representation theory of finite dimensional algebras to construct nice moduli spaces, which can be studied using various powerful techniques based on tilting and silting theory. A particular object of interest is the g-vector fan generated by the classes of 2-term silting complexes, which governs wall-crossing behaviour. Such fans also have an interesting interpretation in physics and enumerative geometry, where they are supposed to form the basis of a scattering diagram, which can be used to determine so called BPS numbers. However, it is often not so straightforward to determine a meaningful fan in this setting, because many of the algebras encountered in geometry and physics are defined over a complete local ring instead of a field. In this talk I will explain how to remedy this using a dimensional reduction argument, which relates the silting theory and stability conditions over the complete local base to a field. As a result, we can again characterise semistable objects using a g-vector fan for algebras of physical or geometric origin, and describe their behaviour under wall-crossing mutation.
Using the geometric model of Opper—Plamondon—Schroll, we investigate the role of gentle algebras in higher homological algebra. We give a characterization of d-cluster tilting subcategories associated to a gentle algebra and show that the examples amount to those one obtains already by studying algebras of Dynkin type A. This reveals a lack of d-abelian and (d+2)-angulated categories arising from gentle algebras, which is surprising due the otherwise rich theory of this class of algebras. The talk is based on joint work with Karin M. Jacobsen and Sibylle Schroll.
Oppermann and Thomas established an intimate relationship between triangulations of even-dimensional cyclic polytopes and the higher (cluster-)tilting theory of higher Auslander algebras of type A in the sense of Iyama. I will describe what is known about the analogous situation when we let the number of vertices of the polytope tend to infinity (which in dimension 2 is an infinity-gon with a single accumulation point). This is a report on joint work with Julian Külshammer.
Generalized associahedra appear throughout mathematics; in particular, they encode much of the combinatorics of cluster algebras of finite type. In this talk I will give a construction of a continuous associahedron motivated by the realization of the generalized associahedron in the physical setting. This continuous associahedron is convex and is related to the theory of cluster algebras. It manifests a cluster theory: the points which correspond to the clusters are on its boundary, and the edges that correspond to mutations are given by intersections of hyperplanes. We will also see that the associahedron shares important properties with the generalized associahedron of type A.
I will report on joint work in progress with Lang Mou. We associate a gentle algebra to each triangulation of any unpunctured surface with orbifold points of order three. The \tau-tilting combinatorics of this gentle algebra coincides with the combinatorics of flips of triangulations. We are able to define Derksen-Weyman-Zelevinsky-like mutations of representations, which is somewhat surprising, since the quivers we consider are allowed to have loops, and the matrix-mutation classes of their skew-symmetrizable matrices may fail to have acyclic representatives. I will explain how this allows us to prove that, whenever one mutates support \tau-tilting pairs, the corresponding Caldero-Chapoton functions obey two different multiplication formulae: a generalized cluster exchange formula if one considers full quiver Grassmannians, and a binomial cluster exchange if one considers only locally-free quiver Grassmannians. The latter provides a proof of a conjecture of Geiss-Leclerc-Schröer in our non-acyclic framework.
A surjective map between commutative noetherian local rings (R,m) -> S is exceptional complete intersection (eci) if its kernel is generated by a regular sequence that is part of a minimal generating set of m. I present two characterizations of eci maps: First, a map is eci if and only if the truncated Atiyah class vanishes at the residue field. This establishes a second characterization in terms of the lattices of thick subcategories of complexes of finite length homology.
Exceptional sequences in module categories over hereditary algebras (e.g. path algebras of quivers) were introduced and studied by W. Crawley-Boevey and C. M. Ringel in the early 1990s, as a way of understanding the structure of such categories. They were motivated by the consideration of exceptional sequences in algebraic geometry by A. I. Bondal, A. L. Gorodontsev and A. N. Rudakov. Exceptional sequences can also be considered over arbitrary finite dimensional algebras, but their behaviour is not so good in general: for example, complete sequences may not exist. We look at different ways of generalising the theory from the hereditary case, with a focus on tau-exceptional sequences, recently introduced in joint work with A. B. Buan (NTNU), motivated by tau-tilting theory, introduced by T. Adachi, O. Iyama and I. Reiten, and the recent definition of signed exceptional sequences in the hereditary case by K. Igusa and G. Todorov.
Stability conditions were introduced by Bridgeland to understand certain constructions in (homological) mirror symmetry. They can be considered as a continuous generalisation of bounded t-structures. Stability conditions form a complex non-compact manifold which has a wall-and-chamber structure whose combinatorics captures interesting aspects of a triangulated category's structure such as its tilting theory.
In this talk, I will describe a partial compactification of the space of stability conditions by allowing "lax" stability conditions with massless objects. For a triangulated category C, a quotient of the space of lax stability conditions, under good conditions, gives a stratified space whose strata are spaces of stability conditions on C/N, where N is a subcategory of massless objects.
I will illustrate the construction using small rank examples.
This is a report on joint work with Nathan Broomhead, David Ploog and Jon Woolf.
The tau-tilting theory of a finite-dimensional algebra naturally leads to the study of projective presentations. These form an extriangulated category, and the classes of its objects in its Grothendieck group can be viewed as a categorification of g-vectors. Interesting geometric objects arise from this point of view, including the g-vector fan and its polytopal realizations.
In this talk, I will introduce another geometric object defined by the vanishing of polynomial equations that reflect the dimension of the extension spaces between projective presentations. We will see how this object is related to g-vectors and F-polynomials, and generalizes known contructions from the cluster algebra world.
This is a report on ongoing work with Nima Arkani-Hamed, Hadleigh Frost, Giulio Salvatori and Hugh Thomas.
Postnikov diagrams are combinatorial objects consisting of strands drawn in a disc, and encode initial seeds for cluster algebra structures on the coordinate ring of the Grassmannian and more general positroid varieties. In particular, such a diagram can be used to compute a cluster of Plücker coordinates, together with its exchange quiver, which then generates the relevant coordinate ring as a cluster algebra. In this talk I will explain joint work with İlke Çanakçı and Alastair King, in which we show that this combinatorial computation is in fact algebraically natural, from the point of view of Jensen, King and Su's Grassmannian cluster category and the representation theory of a pair of algebras attached to the Postnikov diagram.
Tilting theory for exact categories is not properly developed. We give a version generalizing tilting modules for artin algebras to tilting subcategories in arbitrary exact categories. The main difference to before is that in this generality one does not always gets an induced derived equivalence as expected but at least when the category has enough projectives we can characterize when this is the case. We then look examples (e.g. finite dimensional representation of the natural numbers as a poset).
In this talk we will explain when the bounded derived category of a gentle algebra has full exceptional sequences. To every gentle algebra one can associate a dissection into polygons of a compact oriented surface and we describe when an exceptional sequence can be completed to a full exceptional sequence in terms of cuts of the associated surface. Finally, we examine the (extended) braid group action on the set of full exceptional sequences in the case of gentle algebras. This is joint work with Wen Chang and also joint work with Wen Chang and Haibo Jin.
The notion of a torsion class was introduced in the early 1960’s and it has played a central role in the study of abelian categories in general, and module categories of algebras, in particular. A complete classification of functorially torsion classes in the category of finitely presented modules over an artin algebra has been achieved by Adachi, Iyama and Reiten through τ-tilting theory.
In parallel to his work on τ-tilting theory, Iyama set the bases of what we now call d-homological algebra, a theory where the minimal non-split sequences in the category have (d+2)-terms which recovers classical homological algebra when d=1. After Iyama’s work, many classical homological notions were generalised to this new framework. Within this context that Jørgensen introduced d-torsion classes for Jasso’s d-abelian categories.
Recently, it has been shown by Kvamme and, independently, Ebrahimi and Isfahani that every d-abelian category can be seen as a subcategory of a (classical) abelian category. In the first part of the talk we give a characterisation of d-torsion classes in terms of the (classical) torsion classes of the ambient abelian category.
In the second part of the talk, we fix our abelian category to be mod A for an artin algebra A. In this context we show that every functorially finite d-torsion class is generated by a τ_d-rigid object that we construct explicitly and we show some properties of the objects obtained in this way. Time permitting, we give a combinatorial description of all d-torsion classes in the module category of higher Auslander algebras of type A.
This talk is based on joint work with J. Asadollahi, P. Jørgensen and S. Schroll (link), and joint work in progress with J. August, J. Haugland, K. Jacobsen, S. Kvamme and Y. Palu.
Let Q be a quiver. We will talk about a construction of a new quiver Q'=(Q,P) by replacing arrows in Q by a totally ordered poset. Let us call Q' a poset quiver. We consider the path category of Q' possibly with a weakly admissible ideal. Then we study pointwise finite representations. One of our main interests in this object is to study the decomposition of pointwise finite representations. We further analyse some homological properties in this setting. This is a work in progress with Charles Paquette and Job. D. Rock.