Week 2
Program
There will be no banquet in Week 2.
Title and Abstract
Karin Baur (Graz)
Cluster structures for Grassmannians
The category of maximal Cohen-Macaulay modules over a quotient of a preprojective algebra is a cluster category associated to the Scott’s cluster structure on the coordinate ring of the Grassmannian. We study this category, in particular, in the tame cases. We show how to associate SL(k)-friezes to them. This is joint work with Bogdanic-Garcia Elsener and with Faber-Gratz-Serhiyenko-Todorov.
Arkady Berenstein (Oregon)
Noncommutative clusters
The goal of my talk (based on joint work with Vladimir Retakh) is to introduce noncommutative clusters and their mutations, which can be viewed as generalizations of both classical and quantum cluster structures.
A noncommutative cluster X is built on a (torsion-free) group G and a certain collection of its automorphisms. We assign to X a noncommutative algebra A(X) related to the group algebra of G, which is an analogue of the (upper) cluster algebra, and establish a Noncommutative Laurent Phenomenon in some of these algebras.
The class of "cluster groups" G which exhibit the Noncommutative Laurent Phenomenon holds includes: the rank 2 free group and triangular groups of all marked surfaces (they are closely related to the fundamental groups of the ramified double covers). We also expect the Phenomenon to hold for certain noncommutative tori attached to any exchange matrix B.
Thomas Brüstle (Sherbrooke)
Skew-gentle algebras via orbifolds
Skew-gentle algebras are skew group algebras of gentle algebras with an action of order two. Several recent works relate gentle algebras to dissections of marked oriented surfaces, and provide a geometric model for their derived category in terms of curves in the surface. In joint work with Claire Amiot, we use marked surfaces with an automorphism of order two to model skew-gentle algebras and their derived category. We view this as a way towards studying Fukaya categories for orbifolds.
Laurent Demonet (Google / Nagoya)
Combinatorics of mutations and torsion classes [join with O. Iyama, N. Reading, I. Reiten, H. Thomas]
We consider the lattice tors A of torsion classes on a finite dimensional algebra. While this lattice is usually infinite, we show that it can still be well understood by studying its Hasse quiver. Moreover, we give some interpretation this Hasse quiver in terms of A-modules that permits to study some natural quotients of it.
As the Hasse quiver of tors A contains naturally the exchange graph of support tau-tilting modules, tors A can be viewed as a way to extend mutations, even though the behavior at non-functorially finite torsion classes changes drastically. As an example, we will explain how to classify torsion classes over locally gentle algebras [this last part is a join work with A. Chan].
JiaRui Fei (Shanghai Jiao Tong)
Tensor Product Multiplicity via Upper Cluster Algebras
By tensor product multiplicity we mean the multiplicities in the tensor product of any two finite-dimensional irreducible representations of a simply connected Lie group. Finding their polyhedral models is a long-standing problem. The problem asks to express the multiplicity as the number of lattice points in some convex polytope.
Accumulating from the works of Gelfand, Berenstein and Zelevinsky since 1970's, around 1999 Knutson and Tao invented their hive model for the type $A$ cases, which led to the solution of the saturation conjecture. Outside type $A$, Berenstein and Zelevinsky's models are still the only known polyhedral models up to now. Those models lose a few nice features of the hive model.
In this talk, I will explain how to use upper cluster algebras to discover new polyhedral models for all Dynkin types. Those new models improve the ones of Berenstein-Zelevinsky's, or in some sense generalize the hive model.
It turns out that the quivers of relevant upper cluster algebras are related to the Auslander-Reiten theory of presentations, which can be viewed as a categorification of these quivers. The upper cluster algebras are graded by triple dominant weights, and the dimension of each graded component counts the corresponding tensor multiplicity.
The proof also invokes another categorification -- Derksen-Weyman-Zelevinsky's quiver-with-potential model for the cluster algebra. The bases of these upper cluster algebras are parametrized by mu-supported g-vectors. The polytopes will be described via stability conditions.
The talk is based on the preprint arXiv:1603.02521.
Christof Geiss (UNAM)
Rigid representations and Schur roots
This is a report on joint work with B. Leclerc and Jan Schröer. Motivated by the results about finite Dynkin types for our algebras $H=H_F(C,D,\Omega)$ we hope to find a complex geometric interpretation of F-polynomials for all acyclic cluster algebras. A first step into this direction is the following result:
a) The locally free, rigid indecomposable H-modules are parametrized via their rank vector by the real Schur roots associated to $(C,\Omega)$.
b) If M is a locally free rigid indecomposable H-module, then $S:=End_H(M)\cong F[z]/(z^m)$, and M is free as an S-module.
c) The left finite H-bricks are parametrized, via their dimension vector, by the real Schur roots associated to $(C^T,\Omega)$.
For the proof we consider the $F[[z]]$-order $\lim_k H(C,kD,\Omega)$, which allows us to relate H with a species of type $(C,D,\Omega)$ over the field $F((z))$ of Laurent series.
Min Huang (Sherbrooke)
An expansion formula for quantum cluster algebras from unpunctured surfaces
We generalize the expansion formula of Musiker-Schiffler-Williams to the quantum cluster algebras from unpunctured surfaces. Some examples will be given if time permits.
David Hernandez (Paris 7)
Stable maps, category O and categorified exchange relations
We construct stable maps on tensor products of representations in the category O of the Borel subalgebra of an untwisted quantum affine algebra. The construction is based on the study of the action of the Drinfeld-Cartan subalgebra. In A,D,E-cases, these stable maps generalize Maulik-Okounkov K-theoretic stable maps defined for finite-dimensional standard modules using quiver varieties. Our study includes prefundamental representations associated to Baxter's Q-operators.
As an application, we obtain new R-matrices in the category O and categorified exchange relations for the corresponding cluster algebra structures.
Kiyoshi Igusa (Brandeis)
Frieze varieties are mutation invariant
Report on joint paper with Ralf Schiffler with additional comments from Gordana Todorov.
Peter Jorgensen (Newcastle)
c-vectors of 2-Calabi-Yau categories (report on joint work with Milen Yakimov)
We develop a general framework for c-vectors of 2-Calabi-Yau categories with respect to arbitrary cluster tilting subcategories, based on Dehy and Keller's treatment of g-vectors. This approach deals with cluster tilting subcategories which are in general unreachable from each other, and does not rely on (finite or infinite) sequences of mutations. We propose a general program for decomposing sets of c-vectors and identifying each piece with a root system.
Yuki Kanakubo (Sophia)
Cluster theory on double Bruhat cells and crystal bases
For a simply connected, connected, complex simple algebraic group G and Weyl group elements u,v, it is known that the coordinate ring of double Bruhat cell G^{u,v} has a cluster algebra structure. On the other hand, the cell G^{u,e} has a structure of geometric crystal, which is a geometric analog of notion of crystals. In particular, in the case u is the longest element w_0, we can get crystal bases from the cell G^{w_0,e} via a functor called tropicalization and a regular function on G^{w_0,e} called decoration or Berenstein-Kazhdan potential.
In this talk, I will show some results on decorations and relations between cluster variables on the cells G^{u,e} and crystal bases. This is a joint work with G.Koshevoy and T.Nakashima.
Masaki Kashiwara (Kyoto)
Monoidal categorification of the cluster structure on the quantum coordinate ring
The quiver Hecke algebra categorifies the quantum coordinate ring $A_q(\mathfrak{n})$. Similarly, a monoidal subcategory $C_w$ of the module category over the quiver Hecke algebra categorifies the subalgebra $A_q(n(w))$ associated with a Weyl group element $w$. We can prove that it is a monoidal categorification of the cluster algebra structure on $A_q(n(w))$. By using generalized Schur-Weyl functor, we can prove that the Grothendieck category of the module category of quantum affine algebra of type $A^(1)_n$, $A^(2)_n$ and $B^(1)_n$ have a cluster algebra structure and its cluster monomials corresponds to simple modules. This is a joint work with Seok-Jin Kang, Myungho Kim, Se-jin Oh and Euiyong Park.
Yoshiyuki Kimura (Osaka Prefecture)
Twist automorphisms on quantum unipotent cells and the dual canonical bases
Quantum unipotent cell is introduced by De concini-Procesi as a quantum analogue of the coordinate ring of unipotent cells and they proved an isomorphism between quantum analogue of coordinate ring of intersection of unipotent subgroup and shifted Gaussian cells in finite type. In this talk, we construct quantum analogue of twist automorphism whose classical counterpart is introduced by Berenstein-Zelevinsky in the study of total positivity for Schubert varieties. We prove the quantum twist automorphism preserves the dual canonical basis of quantum unipotent cells and quantum cluster monomials. This is a joint work with Hironori Oya.
Alastair King (Bath)
Cluster exchange groupoids and decorated marked surfaces
I will explain how to turn the cluster exchange graph of triangulations of a marked surface into a groupoid in such way that the universal cover can be constructed using `decorated' triangulations. As an application, the space of Bridgeland stability conditions for the associated Ginzburg algebra is shown to be simply connected. (This is joint work with Yu Qiu.)
Daniel Labardini-Fragoso (UNAM)
Generic bases of surface cluster algebras with coefficients
In the last 15 years there has been a lot of research aimed at finding and understanding various types of bases for cluster algebras. One of the proposed bases is the “generic basis”, shown recently by Fan Qin to be indeed a basis for injective reachable upper cluster algebras with full-rank extended exchange matrices. In this talk, based on joint work with Christof Geiss and Jan Schröer, I will sketch an elementary proof of the linear independence of the generic basis for cluster algebras arising, with arbitrary geometric coefficients (not necessarily of full rank), from punctured surfaces with non-empty boundary. I shall also sketch our strategy to show that the generic basis generates the (upper) cluster algebra (more accurately, the Caldero-Chapoton algebra, which in general sits between the cluster algebra and the upper cluster algebra)
Fang Li (Zhejiang)
Unistructurality of cluster algebras
In this talk, we will show the proof for the result that any skew-symmetrizable cluster algebra is unistructural, which was a conjecture by Assem, Schiffler, and Shramchenko. As a corollary, we obtain that a cluster automorphism of a cluster algebra A(S) is just an automorphism of the ambient field F which restricts to a permutation of cluster variables of A(S).
Pierre-Guy Plamondon (Paris 11)
Multiplications formulas for algebras arising from symmetrizable Cartan matrices
One of the challenges of additive categorification of cluster algebras is finding a framework allowing for the study of all skew-symmetrizable cluster algebras. In recent years, C.Geiss, B.Leclerc and J.Schröer have proposed an approach involving algebras defined by quivers with loops and relations. They prove that, in Dynkin cases, these algebras categorify the corresponding cluster algebras. In this talk, I will present a multiplication formula for cluster characters for these algebras. I will also define a cluster category for theses algebras, and show that it possesses many of the homological properties desirable for categorification. (This is joint work with Yann Palu).
Matthew Pressland (Stuttgart)
The Caldero-Chapoton formula as a dimer partition function
Certain elements of a Grassmannian cluster algebra, the twisted Plücker coordinates, are expressible as 'dimer partition functions', i.e. as weighted sums over the set of perfect matchings of a bipartite graph, or dimer model, in the disk, via a formula due to Marsh and Scott. Using the categorification of this cluster algebra by Jensen, King and Su, the more familiar Caldero-Chapton formula provides another expression for the twisted Plücker coordinates, this time in terms of homological data in the category. I will explain how these two formulas are essentially the same, and in so doing relate combinatorial data from the dimer model to homological data in the JKS categorification. For example, a perfect matching on the dimer model encodes a module for the corresponding Jacobian algebra. This is joint work with İlke Çanakçı and Alastair King.
Fan Qin (Shanghai Jiao Tong)
Bases for upper cluster algebras and tropical points
It is known that many (upper) cluster algebras possess very different good bases which are parametrized by the tropical points of Langlands dual cluster varieties. For any given injective reachable upper cluster algebra, we describe all of its bases parametrized by the tropical points. In addition, we obtain the existence of the generic bases for such upper cluster algebras. Our results apply to many cluster algebras arising from representation theory, including quantized enveloping algebras, quantum affine algebras, double Bruthat cells, etc.
Ralf Schiffler (Connecticut)
Frieze varieties: a characterization of the finite-tame-wild trichotomy
This is a joint work with Kyungyong Lee, Li Li, Matt Mills and Alexandra Seceleanu. We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. We give the following new characterization of the finite--tame--wild trichotomy for acyclic quivers in terms of their frieze varieties. An acyclic quiver is representation finite, tame, or wild, respectively, if and only if the dimension of its frieze variety is 0,1, or >1, respectively.
Jan Schröer (Bonn)
Algebras associated with a Cartan datum
Given a Cartan datum (consisting of a symmetrizable generalized Cartan matrix, a symmetrizer and an orientation), we define several classes of algebras, whose representation theory has a lot in common with the representation theory of hereditary algebras. We expect that these algebras yield categorifications of all skew-symmetrizable acyclic cluster algebras. Up to now we can prove this for the Dynkin cases. This is joint work with Bernard Leclerc and Christof Geiss.
Hugh Thomas (New Brunswick)
Newton polytopes of F-polynomials
I will explain a construction of Newton polytopes of F-polynomials of finite type cluster algebras. This construction originated in a construction of the associahedron for linearly oriented type A_n by the physicists Arkani-Hamed, Bai, He, and Yan, as part of their work on scattering amplitudes. It was generalized to finite type cluster algebras starting from acyclic seeds by Bazier-Matte, Douville, Mousavand, Yıldırım, and myself, and then further extended to allow for arbitrary (finite type) starting seeds by Bazier-Matte and Douville. I will also explain a more general conjectural description of Newton polytopes of F-polynomials.
Gordana Todorov (Northeastern)
Torsion Theories and Kappa Map
Joint with: Emily Barnard and Shijie Zhu
We give a representation theoretic interpretation of the combinatorial notion of kappa map as a map on the lattice of torsion classes; in the hereditary case we relate it to Ringel's epsilon map and also to Auslander-Reiten translation.
Milen Yakimov (Louisiana State)
Integral quantum cluster structures
We will describe a general theorem for constructing integral quantum cluster algebras over Z[q^{\pm 1/2}], namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras will then be shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over Z[q^{\pm 1/2}]. As an application, we will show that for every symmetrizable Kac-Moody algebra g and Weyl group element w, the dual canonical form of the corresponding quantum unipotent cell is isomorphic to a quantum cluster algebra (when the base ring is extended to Z[q^{\pm 1/2}]), and that this quantum cluster algebra coincides with the corresponding upper quantum cluster algebra, again defined over Z[q^{\pm 1/2}]. This is a joint work with Ken Goodearl (UCSB).
Bin Zhu (Tsinghua)
Two-term relative cluster tilting subcategories,tau-tilting modules and silting subcategories
Let C be a triangulated category with shift functor [1] and R a rigid subcategory of C. We introduce the notions of two-term R[1]-rigid subcategories, two-term (weak) R[1]-cluster tilting subcategories and two-term maximal R[1]-rigid subcategories. Our main result shows that there exists a bijection between the set of two-term R[1]-rigid subcategories of C and the set of tau-rigid subcategories of mod R, which induces a one-to-one correspondence between the set of two-term weak R[1]-cluster tilting subcategories of C and the set of support tau-tilting subcategories of mod R. This generalizes the our previous results jointed by Yang and Zhou, where R is a cluster tilting subcategory. When R is a silting subcategory, we prove that the two-term weak R[1]-cluster tilting subcategories are precisely two-term silting subcategories. Thus the bijection above induces the bijection given by Iyama-Jorgensen-Yang. This is a joint work with Panyue Zhou.
(last updated: June 11 , 2019)