Lecturer

Se-jin Oh (Sungkyunkwan University)

Quantum tori associated with sequences and their applications

The quantum cluster algebra is introduced by Berenstein-Fomin-Zelevinsky to provide an algebraic-combinatorial framework for investigating the basis of quantum groups. By the celebrated quantum Laurent phenomena, it is a $\mathbb{Z}[q^{\pm 1/2}]$-subalgebra of quantum tori associated with each quantum seed.  In this series of talks, we consider the quantum tori associated with sequences of indices of quantum groups. These quantum tori are deeply related with the root system and (monoidal) categorifications via quiver Hecke algebras.

In the first and second talks, we explore the construction of these quantum tori and their associated applications by focusing on “reduced” sequences. In the third talk, we broaden our scope by removing the constraints of reducedness, discussing recent developments associated with such sequences.

Speakers

Ryota Akagi (Nagoya University)

Explicit forms in lower degrees of rank 2 cluster scattering diagrams

Cluster scattering diagrams, introduced by Gross, Hacking, Keel, and Kontsevich, are a special type of scattering diagrams that encodes all information of cluster patterns, in particular, $c,g$-vectors and $F$-polynomials. Cluster scattering diagrams are a set of walls. Under some assumptions, every $g$-vector fan, which is an object of cluster algebras, can be embedded into a cluster scattering diagram. Moreover, every wall appearing in the $g$-vector fan can be described by using cluster algebras. On the other hand, the structure in the complement of the $g$-vector fan seems to be very complicated. In cluster scattering diagrams, the complement of a $g$-fan is called the Badlands, and one of the most important themes is to clarify all walls in the Badlands. In this talk, I will explain a method to obtain an expression of cluster scattering diagrams explicitly in lower degrees.

Yasuaki Gyoda (The University of Tokyo)

Generalized Markov number and generalized cluster algebra

The Markov equation $x^2+y^2+z^2=3xyz$, studied in the theory of Diophantine approximation by Andrei Markov around 1880, is known to have all positive integer solutions obtained through the operation of mutation in a certain cluster algebra, starting from the trivial solution $(1,1,1)$. In 2023, a generalization of the Markov equation, denoted as $x^2+y^2+z^2+k(yz+zx+xy)=(3+3k)xyz$, which can similarly yield all positive integer solutions through the mutation operation in a certain (generalized) cluster algebra, was discovered by myself and Matsushita. In this presentation, I will introduce recent results about this extension. This presentation includes the contents of joint works with Kodai Matsushita, and with Shuhei Maruyama.

Rei Inoue (Chiba University)

Quantum cluster mutations and 3D integrablity

The tetrahedron equation was proposed by Zamolodchikov in 1980, and the 3D reflection equation was proposed by Isaev and Kulish in 1997, to describe the integrability of 3D lattice models. In this talk I introduce new solutions to these equations by using quantum cluster mutations, based on joint works with Atsuo Kuniba, Xiaoyue Sun, Yuji Terashima and Junya Yagi. We develop the method introduced by Sun and Yagi [arXiv:2211.10702], where they attach three types of quivers to wiring diagrams, and for each of them the quantum mutations yield the solution of the tetrahedron equation in terms of quantum dilogarithm functions. Our strategy is to embed the noncommuting algebra generated by quantum y-variables in the q-Weyl algebra associated with wiring diagrams. We construct adjoint operators R and K that realize Yang-Baxter transformation and reflection transformation respectively.

Yuma Mizuno (Chiba University)

Cluster structures on $q$-Painlevé systems via toric geometry

I will review a geometric interpretation of cluster varieties in terms of blowups of toric varieties discovered by Gross, Hacking, and Keel. I will show that this provides the cluster theoretic interpretation for the geometry of $q$-Painlevé systems studied by Sakai, when the rank (in the sense of linear algebras) of an exchange matrix is two.

Kaveh Mousavand (Okinawa Institute of Science and Technology)

A $\tau$-tilting counterpart of cluster algebras of minimal infinite type

Inspired by the tilting theory of Brenner-Butler, and cluster algebras of Fomin-Zelevinsky, in their seminal work Adachi-Iyama-Reiten introduced the $\tau$-tilting theory to generalize the classical tilting theory from the viewpoint of mutation. In fact, the (in)finiteness of the mutation graph has been decisive in tilting theory, cluster algebras, and more recently in $\tau$-tilting theory. In the early age of cluster algebras, Fomin-Zelevinsky fully classified the cluster algebras of finite mutation type. Following that, Seven, and later Buan-Reiten-Seven, classified the cluster quivers of minimal infinite type. In contrast, there is no full classification of those algebras which are of finite type with respect to mutation in the setting of tilting theory or $\tau$-tilting theory.

This talk is based on my ongoing joint work with Charles Paquette, where we study a new family of algebras, so-called minimal ($\tau$-)tilting infinite algebras, which can be seen as a modern counterpart of cluster algebras of minimal infinite type. Thanks to an elegant result of Demonet-Iyama-Jasso, we can adopt a less technical approach to study the minimal ($\tau$-)tilting infinite algebras. In fact, for the most part I avoid the more technical setting of ($\tau$-)tilting theory and instead treat the behavior of a special subset of indecomposables, known as bricks. I also discuss the importance of minimal ($\tau$-)tilting infinite algebras in our open conjectures, so-called brick-Brauer-Thrall conjectures, and share some new results on our reductive techniques.

Yusuke Nakajima (Kyoto Sangyo University)

Mutations and wall-crossings for dimer models associated to toric cDV singularities

A dimer model, which is a bipartite graph described on the real two-torus, provides the quiver with relations as its dual graph. It is known that for any three-dimensional Gorenstein toric singularity, there exists a dimer model such that the center of the associated path algebra is isomorphic to such a toric singularity, and mutations of dimer models produce several dimer models giving rise to the same toric singularity. On the other hand, for the quiver with relations obtained as the dual of a dimer model, we can consider the stability condition in the sense of A. King. The space of stability conditions has the wall-and-chamber structure and a wall-crossing in the space plays an important role in birational geometry of the associated toric singularity. In this talk, I will focus on a dimer model associated to a toric cDV (compound Du Val) singularity, and discuss a relationship among mutations of dimer models and wall-crossings in the stability space.

Katsuyuki Naoi (Tokyo University of Agriculture and Technology)

Strong duality data of type $A$ and extended $T$-systems

A strong duality datum is a family of finite-dimensional simple modules over a quantum affine algebra $U_q’(\hat{\mathfrak{g}})$. One important property is that we can associate from it the quantum Schur–Weyl duality functor preserving simple modules. Extended $T$-systems, introduced by Mukhin–Young, are short exact sequences satisfied by the tensor product of finite-dimensional simple $U_q’(\hat{\mathfrak{g}})$-modules in types $A$ and $B$ (called snake modules), which contains the celebrated $T$-systems. In this talk, motivated by the generalization of $T$-systems given by Kashiwara–Kim–Oh–Park, we give a generalization of extended $T$-systems for a general strong duality datum of type $A$, which coincides with the Mukhin–Young’s extended $T$-systems when we take a strong duality datum consisting of fundamental modules.

Ryo Takenaka (Osaka Metropolitan University)

On exponents associated with Y-systems

Let $(X_n,\ell)$ be a finite type Dynkin diagram and $\ell$ be a positive integer such that $\ell\geq2$. For a pair $(X_n,\ell)$, we have a system of algebraic relations called Y-system. Solutions of these Y-systems have the periodicity. Due to this fact, Mizuno introduced the notion called exponent. He gave conjectural formulas on exponents and proved for pairs $(A_1,\ell)$ and $(A_n,2)$. In this talk, we assume that $\ell=2$ and show for types $B_n$ and $D_n$.

Wataru Yuasa (Kyoto University)

Skein and cluster algebras with coefficients for unpunctured surfaces

We propose a skein model of the quantum cluster algebras of surface type with coefficients. The skein algebra generalizes Muller’s skein algebra by introducing the wall system, a collection of labeled curves, to the underlying surface. A natural map between skein algebras of walled surfaces obtained by resolving a crossing of walls gives a quasi-homomorphism as defined by Fraser. We also introduce the stated skein algebra of a walled surface and generalizations of wall systems. This talk is based on joint work with Shunsuke Kano and Tsukasa Ishibashi.

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