Related papers: Surface cluster algebra expansion formulae via loo…

In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce the notion of cluster equivalence for these algebras, based on whether their generalized cluster categories are equivalent. We are…

We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…

In this paper, we use subword complexes to provide a uniform approach to finite type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called…

We present an effective method for recovering the topology of a bordered oriented surface with marked points from its cluster algebra. The information is extracted from the maximal triangulations of the surface, those that have exchange…

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…

We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n) in any dimension, and so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic…

Inspired by recent work of Geiss-Leclerc-Schroer, we use Hom-finite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by…

In this work, we introduce a new class of algebras called skew-Brauer graph algebras, which generalize the well-known Brauer graph algebras. We establish that skew-Brauer graph algebras are symmetric and can be defined using a Brauer graph…

Galaxy cluster mass halos ("clusters") in a dark matter simulation are matched to nodes in several different cosmic webs found using the Disperse cosmic web finder. The webs have different simulation smoothings and Disperse parameter…

We obtain a presentation of Schur algebras (and q-Schur algebras) by generators and relations which is compatible with the usual presentation of the enveloping algebra (quantized enveloping algebra) corresponding to the Lie algebra gl(n) of…

We show that in case a cluster algebra coincides with its upper cluster algebra and the cluster algebra admits a grading with finite dimensional homogeneous components, the corresponding Berenstein-Zelevinsky quantum cluster algebra can be…

Expander decompositions of graphs have significantly advanced the understanding of many classical graph problems and led to numerous fundamental theoretical results. However, their adoption in practice has been hindered due to their…

We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings.…

The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…

This is the compendium of the cluster algebra and quiver package for sage. The purpose of this package is to provide a platform to work with cluster algebras in graduate courses and to further develop the theory by working on examples, by…

We present combinatorial approximation algorithms for the weighted correlation clustering problem. In this problem, we have a set of vertices and two weight values for each pair of vertices, denoting their difference and similarity. The…

This paper is a slightly extended version of the talk I gave at the Open Problems in Algebraic Combinatorics conference at the University of Minnesota in May 2022. We introduce two strict order relations on lattice paths and formulate…

Relationship between agents can be conveniently represented by graphs. When these relationships have different modalities, they are better modelled by multilayer graphs where each layer is associated with one modality. Such graphs arise…

Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type.…

Starting from the operator algebra of the (1+1)D Ising model on a spatial lattice, this paper explicitly constructs a subalgebra of smooth operators that are natural candidates for continuum fields in the scaling limit. At the critical…