arXiv subjects

Shunsuke Kano

Publications and source records attributed to Shunsuke Kano.

10 recordsLinked to original sources

arXiv2024

A characterization of pseudo-Anosov mapping classes via tropical cluster transformations

We give a characterization of generic pseudo-Anosov mapping classes purely in terms of their expressions in the shear coordinates, thus giving an answer to a problem raised by Papadopoulos--Penner [PP93]. This characterization has a cluster algebraic generalization called the sign stability introduced in [IK21]. By combining with the results in [IK21], we see that the algebraic entropies of the cluster $\mathcal{A}$- and $\mathcal{X}$-transformations induced by a generic pseudo-Anosov mapping class both coincide with its topological entropy.

math.GT↗

arXiv2024

Earthquake theorem for cluster algebras of finite type

We introduce a cluster algebraic generalization of Thurston's earthquake map for the cluster algebras of finite type, which we call the \emph{cluster earthquake map}. It is defined by gluing exponential maps, which is modeled after the earthquakes along ideal arcs. We prove an analogue of the earthquake theorem, which states that the cluster earthquake map gives a homeomorphism between the spaces of $\mathbb{R}^\mathrm{trop}$- and $\mathbb{R}_{>0}$-valued points of the cluster $\mathcal{X}$-variety. For those of type $A_n$ and $D_n$, the cluster earthquake map indeed recovers the earthquake maps for marked disks and once-punctured marked disks, respectively. Moreover, we investigate certain asymptotic behaviors of the cluster earthquake map, which give rise to "continuous deformations" of the Fock--Goncharov fan.

math.GT↗

arXiv2024

Skein and cluster algebras with coefficients for unpunctured surfaces

We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{Σ,\mathbb{W}}^{A}$ of a walled surface $(Σ,\mathbb{W})$, and prove that it has a quantum cluster structure. The walled surfaces naturally generalize the marked surfaces with multi-laminations, which have been used to describe the quantum cluster algebras of geometric type for marked surfaces by Fomin--Thurston [FT18]. Moreover, we give skein theoretic interpretation for some of quasi-homomorphisms [Fra16] between these quantum cluster algebras.

math.GT↗

arXiv2024

Unbounded $\mathfrak{sl}_3$-laminations around punctures

We continue to study the unbounded $\mathfrak{sl}_3$-laminations [IK22], with a focus on their structures at punctures. A key ingredient is their relation to the root data of $\mathfrak{sl}_3$. After giving a classification of signed $\mathfrak{sl}_3$-webs around a puncture, we describe the tropicalization of the Goncharov--Shen's Weyl group action in detail. We also clarify the relationship with several other approaches by Shen--Sun--Weng [SSW23] and Fraser--Pylyavskyy [FP21]. Finally, we discuss a formulation of unbounded $\mathfrak{g}$-laminations for a general semisimple Lie algebra $\mathfrak{g}$ in brief.

math.RT↗

arXiv2024

Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates

Generalizing the work of Fock--Goncharov on rational unbounded laminations, we give a geometric model of the tropical points of the cluster variety $\mathcal{X}_{\mathfrak{sl}_3,Σ}$, which we call unbounded $\mathfrak{sl}_3$-laminations, based on the Kuperberg's $\mathfrak{sl}_3$-webs. We introduce their tropical cluster coordinates as an $\mathfrak{sl}_3$-analogue of the Thurston's shear coordinates associated with any ideal triangulation. As a tropical analogue of gluing morphisms among the moduli spaces $\mathcal{P}_{PGL_3,Σ}$ of Goncharov--Shen, we describe a geometric gluing procedure of unbounded $\mathfrak{sl}_3$-laminations with pinnings via ``shearings''. We also investigate a relation to the graphical basis of the $\mathfrak{sl}_3$-skein algebra [IY23], which conjecturally leads to a quantum duality map.

math.GT↗

arXiv2024

Entropy of cluster DT transformations and the finite-tame-wild trichotomy of acyclic quivers

The cluster algebra associated with an acyclic quiver has a special mutation loop $τ$, called the cluster Donaldson--Thomas (DT) transformation, related to the Auslander--Reiten translation. In this paper, we characterize the finite-tame-wild trichotomy for acyclic quivers by the sign stability of $τ$ introduced in [IK21] and its cluster stretch factor. As an application, we compute several kinds of entropies of $τ$ and other mutation loops. In particular, we show that the algebraic and categorical entropies of $τ$ are commonly given by the logarithm of the spectral radius of the Coxeter matrix associated with the quiver, and that any mutation loop of finite or tame acyclic quivers have zero algebraic entropy.

math.RT↗

arXiv2023

Train track combinatorics and cluster algebras

The concepts of train track was introduced by W. P. Thurston to study the measured foliations/laminations and the pseudo-Anosov mapping classes on a surface. In this paper, we translate some concepts of train tracks into the language of cluster algebras using the Goncharov--Shen's potential function [GS15]. Through this translation, we prove the sign stability [IK21] of the general pseudo-Anosov mapping classes.

math.GT↗

arXiv2021

Sign stability of mapping classes on marked surfaces II: general case via reductions

We give a cluster algebraic description of the reduction procedure of mapping classes along a multicurve. Based on this description, we characterize pseudo-Anosov mapping classes on a general marked surface in terms of a weaker version of the uniform sign stability, generalizing the main result in the previous paper [IK20]. Moreover we axiomatize a general reduction procedure of mutation loops parametrized by a rational polyhedral cone in the tropical cluster $\mathcal{X}$-variety, which includes both the reduction along a multicurve and the cluster reduction introduced in [Ish19].

math.GT↗

arXiv2021

Categorical dynamical systems arising from sign-stable mutation loops

We give an autoequivalence of the derived category of the Ginzburg dg algebra for a mutation loop satisfying the sign stability introduced in [IK21]. We compute the categorical entropies of their restrictions to some subcategories and conclude that they are both given by the logarithm of the cluster stretch factor. Moreover, we discuss the pseudo-Anosovness of them in the sense of [FFHKL19] and [DHKK14].

math.CT↗

arXiv2021

Algebraic entropy of sign-stable mutation loops

We introduce a property of mutation loops, called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropical $\mathcal{X}$-transformation. A sign-stable mutation loop has a numerical invariant which we call the cluster stretch factor, in analogy with that of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster $\mathcal{A}$- and $\mathcal{X}$-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor.

math.DS↗