On Tropical Friezes Associated with Dynkin Diagrams
نویسنده
چکیده
Tropical friezes are the tropical analogues of Coxeter-Conway frieze patterns. In this note, we study them using triangulated categories. A tropical frieze on a 2-Calabi-Yau triangulated category C is a function satisfying a certain addition formula. We show that when C is the cluster category of a Dynkin quiver, the tropical friezes on C are in bijection with the n-tuples in Z, any tropical frieze f on C is of a special form, and there exists a cluster-tilting object such that f simultaneously takes non-negative values or non-positive values on all its indecomposable direct summands. Using similar techniques, we give a proof of a conjecture of Ringel for cluster-additive functions on stable translation quivers.
منابع مشابه
To my dearest parents and to
This thesis is concerned with higher cluster tilting objects in generalized higher cluster categories and tropical friezes associated with Dynkin diagrams. The generalized cluster category arising from a suitable 3-Calabi-Yau differential graded algebra was introduced by C. Amiot. It is Hom-finite, 2-Calabi-Yau and admits a canonical cluster-tilting object. In this thesis, we extend these resul...
متن کاملGeneralised Friezes and a Modified Caldero-chapoton Map Depending on a Rigid Object
The (usual) Caldero-Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps “reachable” indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalises the idea that the cluster category is a “categorification” of the cluster algebra. The definition of the Caldero-C...
متن کاملDynkin Diagrams of Cp
We investigate N = 2 supersymmetric sigma model orbifolds of the sphere in the large radius limit. These correspond to N = 2 superconformal field theories. Using the equations of topological-anti-topological fusion for the topological orbifold, we compute the generalized Dynkin diagrams of these theories i.e., the soliton spectrum which was used in the classification of massive superconformal t...
متن کاملPhylogenetic trees and the tropical geometry of flag varieties
We will discuss some recent theorems relating the space of weighted phylogenetic trees to the tropical varieties of each flag variety of type A. We will also discuss the tropicalizations of the functions corresponding to semi-standard tableaux, in particular we relate them to familiar functions from phylogenetics. We close with some remarks on the generalization of these results to the tropical...
متن کاملSolvable lattice models labelled by Dynkin diagrams
An equivalence between generalised restricted solid-on-solid (RSOS) models, associated with sets of graphs, and multi-colour loop models is established. As an application we consider solvable loop models and in this way obtain new solvable families of critical RSOS models. These families can all be classified by the Dynkin diagrams of the simply-laced Lie algebras. For one of the RSOS models, l...
متن کامل