相关论文: Cluster scattering coefficients in rank 2

The purpose of this paper is to translate the expression of rank 2 cluster scattering diagrams via dilogarithm elements into via formal power series. As a corollary, we prove some conjectures introduced by Thomas Elgin, Nathan Reading, and…

In this paper, we prove the consistency relations of rank $2$ cluster scattering diagrams of affine type by using the pentagon relation.

We study rank-2 cluster scattering diagrams through moduli spaces of quiver representations and a recently developed combinatorial framework of tight gradings. Combining quiver-theoretic and combinatorial methods, we prove and extend a…

Scattering diagrams arose in the context of mirror symmetry, but a special class of scattering diagrams (the cluster scattering diagrams) were recently developed to prove key structural results on cluster algebras. We use the connection to…

We give more precise statements of Fock-Goncharov duality conjecture for cluster varieties parametrizing ${\rm SL}_{2}/{\rm PGL}_{2}$-local systems on the once punctured torus. Then we prove these statements. Along the way, using distinct…

We review some important results by Gross, Hacking, Keel, and Kontsevich on cluster algebra theory, namely, the column sign-coherence of $C$-matrices and the Laurent positivity, both of which were conjectured by Fomin and Zelevinsky. We…

We study the polygons defining the dominance order on $\mathbf{g}$-vectors in cluster algebras of rank 2.

We provide a complete classification of the singularities of cluster algebras of finite cluster type. This extends our previous work about the case of trivial coefficients. Additionally, we classify the singularities of cluster algebras for…

Clifford indices of vector bundles on algebraic curves were introduced in a previous paper of the authors. In this paper we study bundles of rank 2 which compute these Clifford indices. This is of particular interest in the light of…

In this paper, we study wall elements of rank 2 cluster scattering diagrams based on dilogarithm elements. We derive two major results. First, we give a method to calculate wall elements in lower degrees. By this method, we may see the…

We introduce a new class of combinatorial objects, named tight gradings, which are certain nonnegative integer-valued functions on maximal Dyck paths. Using tight gradings, we derive a manifestly positive formula for any wall-function in a…

This is a first step guide to the theory of cluster algebras. We especially focus on basic notions, techniques, and results concerning seeds, cluster patterns, and cluster algebras.

The random networks enriched with additional structures as metric and group-symmetry in background metric space are investigated. The important quantities like he clustering coefficient as well as the mean degree of separation in such…

We construct scattering diagrams for Chekhov-Shapiro's generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out…

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.

We conjecture a new set of analytic relations for scattering amplitudes in planar N=4 super Yang-Mills theory. They generalise the Steinmann relations and are expressed in terms of the cluster algebras associated to Gr(4,n). In terms of the…

In this note, we give some new families of two-stage spaces for which the torus rank conjecture is affirmed.

The 2-adic valuation of the Stirling numbres is examined. We conjecture pattrens about the distributions of these valuations in residue classes modulo powers of 2.

We consider the mutation invariants of cluster algebras of rank 2. We characterize the mutation invariants of finite type. Two examples are provided for the affine type and we prove the non-existence of Laurent mutation invariants of…

In 2013, Lee, Li, and Zelevinsky introduced combinatorial objects called compatible pairs to construct the greedy bases for rank-2 cluster algebras, consisting of indecomposable positive elements including the cluster monomials.…