Title and Abstract

Sota Asai(Osaka)

The rigid parts of the elements of the real Grothendieck groups

This talk is based on joint work with Osamu Iyama in progress. Let $A$ be a finite-dimensional algebra over a field $K$. Then, in the homotopy category $\mathsf{K}^\mathrm{b}(\mathrm{proj}A)$, any 2-term complex $X$ has the maximum presilting direct summand, which can be called the rigid part of $X$. By using decompositions of 2-term complexes, Derksen-Fei introduced the canonical decompositions in the Grothendieck group $K_0(\mathrm{proj} A)$, so we can define the rigid part of each $\theta \in K_0(\mathrm{proj} A)$ from its canonical decomposition. However, in this definition, it is open whether the rigid part of $m \theta$ is always $m$ times of the rigid part of $\theta$ for any positive integer $m \in {\mathbb{Z}} _{\geq 1}$. To avoid this problem, we define the rigid part of the elements in the Grothendieck group in a different way; namely, we use the real Grothendieck group ${K_0(\mathrm{proj} A)} _\mathbb{R}$ and numerical torsion pairs of Baumann-Kamnitzer-Tingley. In this talk, I will explain the detail of our definition and some important properties of our rigid parts.

Takeru Asaka(Tokyo)

Some calculations about earthquake maps in the cross ratio coordinates

Thurston defined the earthquake map. It is a deformation of hyperbolic surfaces and considered as the bijective map on the universal cover $\mathbb{H}$. The earthquake theorem says that for any orientation-preserving homeomorphism $f$ of $S^1$, there uniquely exists an earthquake extended to $f$. Bonsante—Krasnov—Schlenker extended the theorem to the enhanced Teichmüller space of marked surfaces. We calculate some earthquake maps on marked surfaes by the cross ratio coordinates. They are earthquake maps along ideal arcs and along a simple closed geodesic.

Mikhail Bershtein (Landau Institute, Skoltech, IPMU)

$q$-Painleve equations, mutations and reductions

Goncharov and Kenyon constructed cluster integrable systems corresponding to bipartite graphs on a torus. These systems are invariant with respect to cluster mutations which correspond to mutations of a quadrilateral face. It appears that there is another class of transformations that preserves these integrable systems, namely mutation with respect to zig-zags. This leads to a natural extension of the Goncharov-Kenyon class of cluster integrable systems by their Hamiltonian reductions. In particular this can be applied for the construction of the phase space of the $q$-Painleve equation. Based on joint works with P. Gavrylenko, A. Marshakov, M. Semenyakin

Peigen Cao (U Hong Kong)

Cluster-additive functions and tropical points

Let A be a symmetrizable generalized Cartan matrix of size r. A cluster-additive function associated to A is a map from Z times [1,r] to Z satisfying certain mesh type relations. Such functions were introduced by Ringel, which are closely related with additive functions in representation theory. We prove that cluster-additive functions associated to A are in bijection with the tropical points associated to a skew-symmetrizable matrix defined from A. We give an realization of cluster-additive functions using d-compatibility degree in cluster algebras. As one of the applications, we give a conceptual proof of Ringel’s conjecture on cluster-additive functions. This is based on a work in progress joint with Antoine de St. Germain and Prof. Jiang-Hua Lu.

Naoki Fujita (Kumamoto)

Toric degenerations and Newton-Okounkov bodies of flag varieties arising from cluster structures

A toric degeneration is a flat degeneration of a projective variety to a toric variety, which can be used to apply the theory of toric varieties to other projective varieties. Using a multiplicative property of the dual canonical basis, Caldero constructed a toric degeneration of a flag variety to the toric variety associated with a string polytope. Magee and Bossinger-Fourier (respectively, Genz-Koshevoy-Schumann) studied relations between GHKK (Gross-Hacking-Keel-Kontsevich) superpotential polytopes and string polytopes in type A (respectively, in simply-laced type). In this talk, we define Newton-Okounkov bodies of flag varieties using a cluster structure, and see how these are related to GHKK superpotential polytopes. These Newton-Okounkov bodies give a cluster-theoretic realization of string polytopes in general Lie type. This talk is based on a joint work with Hironori Oya.

Yasuaki Gyoda (Tokyo)

Exchange quiver of root systems, generalized path algebras and cluster algebras

To date, it has been shown that there exists a bijective correspondence between the roots of the Dynkin root system, the indecomposable module of algebra of finite representation type, and the cluster variables of a cluster algebra of finite type, such that the structure known as the “cluster structure” is preserved. This implies the existence of a common combinatorial structure among these three mathematical objects, and the study of this structure is now contributing significantly to the development of these three fields.
The cluster structure introduced in each object can be expressed using an “exchange graph,” a graph whose edges are not oriented. In this presentation, I will introduce a stronger structure by considering an “exchange quiver,” in which each edge of the exchange graph is oriented, and then I will introduce three “stronger structures” by Dynkin root systems, (generalized) path algebras, and cluster algebras of finite type. Moreover, we will show, with concrete examples, that the three “stronger structures” by variables are the same.

Ivan Chi-Ho Ip (HKUST)

Regular Positive Representations

In the study of positive representations of split real quantum groups, it is natural to consider an associated embedding of $U_q(\mathfrak{g})$ to certain quantum cluster algebra related to the moduli space of framed $G$-local systems. Over the past few years, various new variations of such representations are realized as certain reduction of the original one, and share some common features which we call “regular”. We will explain the current status of the classification of these regular positive representations.

Takeyoshi Kogiso (Josai)

I will talk about two kinds of deformations of a Markov equation. One deformation is related to q-deformation of rational numbers introduced by Morier-Genoud and V.Ovsienko , that is connected to knot theory, hyperbolic geometry, cluster algebra. The other deformation is related to Castling transformation of t-dimensional prehomogeneous vector spaces.

Yuya Mizuno (Osaka metropolitan)

Fans and polytopes in tilting theory

For a finite dimensional algebra $A$, the 2-term silting complexes of $A$ gives a simplicial complex $\Delta(A)$, called the $g$-simplicial complex, a polytope $P(A)$, called the $g$-polytope and a fan $\Sigma(A)$, called the $g$-fan. We study several properties of these three objects. In particular, we give tilting theoretic interpretations of the $h$-vectors and Dehn-Sommerville equations of $\Delta(A)$. Moreover, we discuss convexity of $P(A)$ and its dual polytope. We also discuss a classification of rank 2 $g$-fans. This is joint work with Aoki-Higashitani-Iyama-Kase.

Kota Murakami (Kyoto)

Categorifications of deformed symmetrizable generalized Cartan matrices

Motivated from studies of the representation theory of quantum loop algebras, Geiss-Leclerc-Schröer introduced the notion of the generalized preprojective algebra associated with a symmetrizable generalized Cartan matrix and its symmetrizer. We define and study a several parameter deformation of a symmetrizable generalized Cartan matrix, which is a numerical aspect of the multi-graded module category of the generalized preprojective algebra. Under a certain condition, which is satisfied in all the symmetric cases or in all the finite and affine cases, our definition coincides with that of the mass-deformed Cartan matrices introduced by Kimura-Pestun in their study of quiver W-algebras. In particular, we will interpret some numerical formula about this matrix in terms of braid group symmetries of our graded module category. This is a joint work with Ryo Fujita (RIMS).

Yu Qiu (Tsinghua)

Exchange graph of partial cluster tilting objects and subsurface collapsing

We introduce a new class of triangulated categories, which are Verdier quotients between 3-Calabi-Yau categories from (decorated) marked surfaces, and show that its spaces of stability conditions can be identified with moduli spaces of framed quadratic differentials on Riemann surfaces with arbitrary order zeros and arbitrary higher order poles. The skeletons of these spaces are coverings of exchange graphs of partial cluster tilting objects. This is a joint work with Anna Barbieri, Martin Möller and Jeonghoon So.

Takao Suzuki (Kindai)

Cluster algebra and $q$-Painlevé equation: higher order generalization and degeneration structure

Recently, a birational representation of an extended affine Weyl group of type $A_{mn-1}^{(1)}\times A_{m-1}^{(1)}\times A_{m-1}^{(1)}$ was formulated with the aid of cluster mutation by Inoue-Ishibashi-Oya and Masuda-Okubo-Tsuda. We derive Jimbo-Sakai’s $q$-Painlevé VI equation and its higher order generalizations from this group. We also propose confluences of vertices of quivers which imply the degeneration structure of the $q$-Painlevé equations. This talk is based on the joint work with Prof. Naoto Okubo.

Wataru Yuasa (RIMS, OCAMI)

Cluster and Skein algebras of unpunctured surfaces for $\mathfrak{sp}_4$

We introduce a skein algebra consisting of $\mathfrak{sp}_4$-webs on a marked surface $\Sigma$ without punctures and its $\mathbb{Z}_q$-subalgebra called the ``$\mathbb{Z}_q$-form’’. We prove that the boundary-localization of the $\mathbb{Z}_q$-form is included in a quantum cluster algebra that quantizes the function ring of the moduli space of all decorated twisted $Sp_4$-local systems on $\Sigma$. In this talk, we explain the construction of the inclusion and a conjectural characterization of cluster variables using $\mathfrak{sp}_4$-webs. This talk is based on joint work with Tsukasa Ishibashi.

Junya Yagi (Tsinghua)

Cluster transformations, the tetrahedron equation and three-dimensional gauge theories

Solutions of Zamolodchikov’s tetrahedron equation define integrable 3D lattice models in statistical mechanics, just as solutions of the Yang-Baxter equation define integrable 2D lattice models. I will explain how we can construct solutions of the tetrahedron equation using certain quantum cluster transformations. This is based on my joint work with Xiao-yue Sun.

Michihisa Wakui (Kansai)

This talk is based on results in the master thesis written by Koki Oya under my advice at Kansai University in this year. I would like to explain his results for him since he graduates in March.

The concept of partition q-series for quiver mutation loops is introduce by Akishi Kato and Yuji Terashima to obtain significant physical quantities by using only combinatorial data. The partition q-series are related to wide area such as Dnaldson-Thomas theory, low-dimensional topology, representation theory and quantum field theory. In this talk we give several examples of quiver mutation loops and partition q-series associated with affine quivers of type A.