Related papers: Cluster structures and subfans in scattering diagr…
Leclerc constructed a conjectural cluster structure on Richardson varieties in simply laced types using cluster categories. We show that in type A, his conjectural cluster structure is in fact a cluster structure. We do this by comparing…
This paper explores the cluster algebra structure of the moduli space $\mathscr{A}_{\mathrm{SL}_{n+1},\mathbb{S}}$ of twisted $\mathrm{SL}_{n+1}$-local systems on a surface. We derive general recurrence relations for cluster variables…
This work develops a flexible and mathematically sound framework for the design and analysis of graph scattering networks with variable branching ratios and generic functional calculus filters. Spectrally-agnostic stability guarantees for…
Given a certain triangulation of a punctured surface with boundary, we construct a new triangulated surface without punctures which covers it. This new surface is naturally equipped with an action of a group of order two, and its quotient…
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings…
We revisit the total scatterings (in terms of extinction, scattering and absorption cross sections) by arbitrary clusters of nonmagnetic particles that support optically-induced magnetic responses. Our reexamination is conducted from the…
We conjecture a characterization of a cluster automorphism as an algebra homomorphism from the cluster algebra to itself that restricts to a bijection between two clusters. This formulation does not require that the map commutes with…
In previous work, we initiated the study of the cohomology of locally acyclic cluster varieties. In the present work, we show that the mixed Hodge structure and point counts of acyclic cluster varieties are essentially determined by the…
Nearest neighbor (k-NN) graphs are widely used in machine learning and data mining applications, and our aim is to better understand what they reveal about the cluster structure of the unknown underlying distribution of points. Moreover, is…
In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras.…
Though it is generally assumed that massive molecular clouds are the progenitors of globular clusters, their detailed formation mechanism is still unclear. Standard scenarios based on the collapse of a smooth matter distribution suffer from…
Motivated by questions in biological classification, we discuss some elementary combinatorial and computational properties of certain set systems that generalize hierarchies, namely, 'patchworks', 'weak patchworks', 'ample patchworks' and…
We provide a concrete realization of the cluster algebras associated with Q-systems as amalgamations of cluster structures on double Bruhat cells in simple algebraic groups. For nonsimply-laced groups, this provides a cluster-algebraic…
In the framework of the Time Dependent Scattering Theory we discuss three forms of Cluster Separability as well as the conditions for the representation of the scattering system dynamics implied by their respective use.
Motivated by the cluster structure of two-loop scattering amplitudes in N=4 Yang-Mills theory we define "cluster polylogarithm functions". We find that all such functions of weight 4 are made up of a single simple building block associated…
In [22], Crane and Sheppard considered the structure of the Poincare group as a 2-Group, and derived important information about its representations in a 2-Category suited for representations of non-compact 2-groups, following a lead of…
In this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters. We prove that cluster algebras of Dynkin type and cluster algebras of rank 2 are unistructural,…
Fock-Goncharov's moduli spaces $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ of framed ${\rm PGL}_3$-local systems on punctured surfaces $\frak{S}$ provide prominent examples of cluster $\mathscr{X}$-varieties and higher Teichm\"uller spaces. In a…
We study a two-species bidirectional exclusion process, and a single species variant, which is motivated by the motion of organelles and vesicles along microtubules. Specifically, we are interested in the clustering of the particles and…
We first study the (canonical) orbit category of the bounded derived category of finite dimensional representations of a quiver with no infinite path, and we pay more attention on the case where the quiver is of infinite Dynkin type. In…