Title, Abstract & Slides

9/27 Monday

Aaron Chan Categorification of (quasi-)triangulations of unpunctured non-orientable marked surfaces

Abstract: Given an orientable marked surface, one can associate to it a cluster algebra where cluster variables are arcs, clusters are triangulations, and mutations are flips. For the case of a non-orientable marked surface, Teichmuller theory tells us that it is still possible to talk about flip of quasi-triangulations (i.e. maximal collections of arcs and one-sided simple closed curves). Mimicking the cluster structure on orientable marked surfaces, Dupont-Palesi constructed a commutative algebra, called quasi-cluster algebra, associated with a non-orientable marked surface. Since the former can be (additively) categorified, it is natural to ask for the categorification of the latter. In this talk, we will explain the first step in this direction for unpunctured non-orientable surfaces, namely, by categorifying triangulations and their mutations. We will then talk about the key ingredient in our categorification, namely, symmetric representations (in the sense of Derksen-Weyman and Boos-Cerulli Irelli) over rings with (anti-)involution - their yet mysterious cluster/tau-tilting theory is just as interesting in its own right. If time permits, we will also explain the categorification of quasi-triangulations. This talk is based on a joint work in progress with Veronique Bazier-Matte and Kayla Wright.

Takahide Adachi Silting-discrete algebras

Abstract: Silting complexes were introduced by Keller and Vossieck to study bounded t-structures on the derived categories of path algebras. Aihara and Iyama introduced the notion of silting mutation, which is an analog of mutation in cluster algebras. From the viewpoint of silting mutation theory, a silting-discrete algebra is important because it has a property that any silting complex is given by a finite sequence of irreducible silting mutations. In this talk, I will explain some properties of silting-discrete algebras.

Sota Asai Purely non-rigid regions of the Grothendieck groups

Abstract: This talk is based on joint work with Osamu Iyama (the University of Tokyo). Let $A$ be a finite-dimensional algebra over a field $K$. For each 2-term presilting complex $U$ in the homotopy category $\mathrm{K}^\mathrm{b}(\operatorname{proj} A)$, we can associate a cone in the real Grothendieck group $K_0(\operatorname{proj} A)_\mathbb{R}$. It is an interesting problem to determine the complement of the union of these cones. We found a nice description of this complement by using a certain open neighborhood of the cone associated to each 2-term presilting complex and the purely non-rigid region of $K_0(\operatorname{proj} A)_\mathbb{R}$, which we newly introduced. I will talk about this description.

Tsukasa Ishibashi Wilson lines and their Laurent positivity

Abstract: The moduli space of framed G-local systems with pinnings on a marked surface has been introduced by Goncharov-Shen (2019) and shown to have a natural cluster Poisson structure. In this talk, we introduce a new class of G-valued functions, called the Wilson lines, associated with the homotopy classes of arcs between boundary intervals. These functions give morphisms from the moduli space to the algebraic group G, and their matrix coefficients in a finite-dimensional representation of G give rise to Laurent polynomials in every cluster Poisson chart. We show that, for a suitable choice of matrix coefficients, these Laurent polynomials have positive integral coefficients in the cluster Poisson chart associated with any decorated triangulation. If time permits, I’d like to mention the ongoing research on the relation to the skein theory and the quantization of Wilson lines. This talk is based on a joint work with Hironori Oya.

Akishi Ikeda Arcs on surfaces vs modules over algebras

Abstract: There is a class of cluster algebras whose combinatorial structures can be understood through the geometry of surfaces. In this talk, I will attempt to explain this relationship based on the homological mirror symmetry. In particular, I will review the works due to Qiu Yu and Zhou Yu, which give the correspondence between some class of arcs on surfaces and modules over 3-Calab-Yau algebras (or cluster tilting objects in corresponding cluster categories). If times permits, I will introduce my joint work with Qiu Yu and Zhou Yu, which extend the above correspondence to larger class of Calabi-Yau algebras.

Tatsuki Kuwagaki Sheaf quantization and principal cluster variety

Abstract: Sheaf quantization is a Betti version of quantization of Lagrangian submanifolds, which can also be considered as an enhanced version of the notion of constructible sheaf. By the work of Shende—Treumann—Williams—Zaslow, one can study the cluster structures on wild character varieties using certain constructible sheaves. These constructible sheaves admit liftings to sheaf quantizations. On the other hand, cluster structures also admit an enhanced version: principal cluster structures. In this talk, I’ll explain how these two enhanced versions are related. This talk is based on a joint work with Tsukasa Ishibashi.

9/28 Tuesday

Yuma Mizuno q-Painlevé systems on cluster Poisson varieties

Abstract: Sakai introduced a geometric framework for discrete Painlevé systems. According to Sakai’s theory, there are three types of discrete Painlevé systems: elliptic type, multiplicative type, and additive type. A discrete Painlevé system of multiplicative type is also called a q-Painlevé system. In this talk, I will explain Sakai's theory for q-Painlevé systems in terms of cluster theory. The key tool is a toric interpretation of cluster Poisson varieties given by Gross-Hacking-Keel.

Yusuke Nakajima Combinatorial mutations of polytopes arising from plabic graphs

Abstract: It is known that the labeled seed obtained from a certain plabic graph, which is a planar bicolored graph embedded in a disk, gives rise to the cluster structure of the Grassmannian. Using such a plabic graph, we can construct the Newton--Okounkov polytope which gives a toric degeneration of the Grassmannian. In general, there are several plabic graphs giving the cluster structure of the same Grassmannian, and those plabic graphs are related by the mutations. The Newton--Okounkov polytopes associated to such plabic graphs are related by the tropicalized cluster mutations by the work of Rietsch and Williams. On the other hand, there is the other operation called the combinatorial mutation of polytopes, which was introduced in the context of mirror symmetry.

In my talk, I will discuss the relationship between the tropicalized cluster mutation of the Newton--Okounkov polytope associated to a plabic graph and the combinatorial mutation of such a polytope. This talk is based on a joint work with A. Higashitani (arXiv.2107.04264).

Wataru Yuasa Skein realization of cluster algebras with coefficients from marked surfaces

Abstract: The skein algebra is a quotient algebra of the algebra of knot diagrams on a surface modulo skein relations. It has many variants corresponding to types of surfaces. For a marked surface with no punctures, Muller defined a skein algebra of the surface and proved that it is isomorphic to the cluster algebra from the surface. In this talk, we introduce a skein algebra of a marked surface equipped with a wall (= a collection of loops and ideal arcs). For the unpunctured case, we show that the skein algebra of the walled surface realizes the cluster algebra with coefficients from a surface with integral lamination determined by the wall. The coefficients have a description in terms of laminated Teichmuller theory by Fomin-Thurston. This realization is with coefficients version of Muller's work. We also give some observations for the skein algebra of a walled surface with punctures. This talk is based on joint work with Tsukasa Ishibashi and Shunsuke Kano.

Shunsuke Kano Pseudo-Anosov properties in cluster algebras

Abstract: Pseudo-Anosov mapping class is the most general element in the mapping class group of a punctured surface and there are many theories for them. We introduce a new property for mutation loops, called the sign stability, as a cluster algebraic analogy of pseudo-Anosov mapping classes. In this talk, we show that the pseudo-Anosov property is actually equivalent to a kind of sign stability by considering a mapping class as a mutation loop. A sign-stable mutation loop has the stretch factor like pseduo-Anosov mapping classes. We compute the algebraic entropies of cluster transformations and the autoequivalence of derived categories of Ginzburg dg algebras induced by a sign-stable mutation loop, and we see that they are given by the logarithm of the stretch factor. This talk is based on a joint work with Tsukasa Ishibashi.

Naoto Okubo Mutation conbinatorics and q-Painleve systems

Abstract: In this talk, we discuss mutation conbinatorics. We present q-Painleve systems by using cluster mutations. This talk is based on joint work with Tetsu Masuda and Teruhisa Tsuda.

Yasuaki Gyoda Compatibility degree of cluster complexes

Abstract: I will introduce a new function on the set of pairs of cluster variables, called the compatibility degree (of cluster complexes). The compatibility degree which I deal with in this talk is a generalization of the ``classical" compatibility degree introduced by Fomin and Zelevinsky. The classical one defines the generalized associahedron, and it is used to give the classification of cluster algebras of finite type. Cao and Li generalized this degree to that of cluster complexes by using d-vectors. We give another generalization by using f-vectors. This is joint work with Changjian Fu.

9/29 Wednesday

Yuya Mizuno Arc diagrams and 2-term simple-minded collections of preprojective algebras of type A

Abstract: The notion of arc diagrams was introduced by N.Reading. He showed that the set of noncrossing arc diagrams is in bijective correspondence with elements of the symmetric group and it provides a combinatorial model for canonical join representations. In this talk, we give an explicit description of semibricks and 2-term simple-minded collections over preprojective algebras of type A via arc diagrams. In particular, we explain a bijection between the set of noncrossing arc diagrams (resp. the set of double arc diagrams) and the set of semibricks (resp. the set of 2-term simple-minded collections) over the algebra.

Toshiya Yurikusa Bongartz completion via c-vectors

Abstract: Bongartz completion is an important subject in representation theory. In tau-tilting theory, we can characterize it using c-vectors. From this point of view, we introduce the notion of Bongartz completion for cluster algebras using c-vectors, and we give some of its properties and applications. This is a joint work with Peigen Cao and Yasuaki Gyoda.

Naoki Fujita Semi-toric degenerations of Schubert varieties arising from cluster structures on flag varieties

Abstract: A toric degeneration is a flat degeneration into an irreducible normal toric variety. In the case of a flag variety, its toric degeneration with desirable properties induces degenerations of Schubert varieties into unions of irreducible toric subvarieties, called semi-toric degenerations. Semi-toric degenerations of Schubert varieties are closely related to Schubert calculus. For instance, Kogan-Miller constructed semi-toric degenerations of Schubert varieties from Knutson-Miller's semi-toric degenerations of matrix Schubert varieties which give a geometric proof of the pipe dream formula of Schubert polynomials. In this talk, we consider a toric degeneration of a flag variety arising from its cluster structure, and see that it induces semi-toric degenerations of Schubert varieties, which can be regarded as generalizations of Kogan-Miller's semi-toric degeneration.

Hironori Oya Isomorphisms among quantum Grothendieck rings and their applications

Abstract: A quantum Grothendieck ring of the monoidal category of finite-dimensional modules over a quantum loop algebra $U_{q}(L\mathfrak{g})$ is a one parameter deformation of the usual Grothendieck ring. In this talk, I explain several algebra isomorphisms among the quantum Grothendieck rings, and their applications. One remarkable point is that they include the isomorphisms between the quantum Grothendieck ring of a simply-laced type and that of a non-simply-laced type. It leads to new positivity results for the simple $(q, t)$-characters in the non-simply-laced cases, and new methods of calculation of the simple $q$-characters. This talk is based on a joint work with Ryo Fujita, David Hernandez, and Se-jin Oh (arXiv:2101.07489).

Ryo Fujita Deformed Cartan matrices and generalized preprojective algebras of finite type

Abstract: In their study of deformed W-algebras associated with simple Lie algebras, E. Frenkel-Reshetikhin (1998) introduced certain (q,t)-deformations of the Cartan matrices. They play an important role in representation theory of quantum affine algebras. In this talk, we give an interpretation of these (q,t)-deformed Cartan matrices and their inverses in terms of bigraded modules over the generalized preprojective algebras of Langlands dual type in the sense of Geiss-Leclerc-Schröer (2017). As an application, we compute the first extension groups between the generic kernels introduced by Hernandez-Leclerc (2016), and propose a conjecture that their dimensions coincide with the pole orders of the normalized R-matrices between the corresponding Kirillov-Reshetikhin modules. This is a joint work with Kota Murakami.

Kota Murakami PBW parametrizations and generalized preprojective algebras

Abstract: Geiss-Leclerc-Schröer has introduced a class of associative algebras, called generalized preprojective algebras. This class of algebra is defined as a quiver with relations associated with a symmetrizable generalized Cartan matrix and its symmetrizer. In this talk, we review the definition and basic properties of generalized preprojective algebras. Then, we understand some combinatorial information about canonical basis of quantum groups through representation theory of generalized preprojective algebras.

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