## 25 Citations

$\tilde{A}$ and $\tilde{D}$ type cluster algebras: Triangulated surfaces and friezes

- Mathematics
- 2021

By viewing Ã and D̃ type cluster algebras as triangulated surfaces, we find all cluster variables in terms of either (i) the frieze pattern (or bipartite belt) or (ii) the periodic quantities…

ON TROPICAL FRIEZES ASSOCIATED WITH DYNKIN DIAGRAMS 3

- Mathematics
- 2012

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…

FRIEZES, STRINGS AND CLUSTER VARIABLES

- MathematicsGlasgow Mathematical Journal
- 2011

Abstract To any walk in a quiver, we associate a Laurent polynomial. When the walk is the string of a string module over a 2-Calabi–Yau tilted algebra, we prove that this Laurent polynomial coincides…

On tropical friezes associated with Dynkin diagrams

- Mathematics
- 2012

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…

Friezes of type D

- Mathematics
- 2014

In this article, we establish a link between the values of a frieze of type D and some values of a particular frieze of type A. This link allows us to compute, independently of each other, all the…

Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object, II

- Mathematics
- 2016

A Caldero–Chapoton map for infinite clusters

- Mathematics
- 2010

We construct a Caldero-Chapoton map on a triangulated category with a cluster tilting subcategory which may have infinitely many indecomposable objects. The map is not necessarily defined on all…

Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object

- MathematicsNagoya Mathematical Journal
- 2015

Abstract 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…

Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics

- Mathematics, Art
- 2015

Frieze patterns of numbers, introduced in the early 1970s by Coxeter, are currently attracting much interest due to connections with the recent theory of cluster algebras. The present survey aims to…

5 J un 2 01 4 GENERALISED FRIEZES AND A MODIFIED CALDERO-CHAPOTON MAP DEPENDING ON A RIGID OBJECT

- Mathematics
- 2018

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…

## References

SHOWING 1-10 OF 28 REFERENCES

Positivity for Regular Cluster Characters in Acyclic Cluster Algebras

- Mathematics
- 2009

Let $Q$ be an acyclic quiver and let $\mathcal A(Q)$ be the corresponding cluster algebra. Let $H$ be the path algebra of $Q$ over an algebraically closed field and let $M$ be an indecomposable…

Generalized Chebyshev Polynomials and Positivity for Regular Cluster Characters

- Mathematics
- 2009

Let $Q$ be an acyclic quiver and let $\mathcal A(Q)$ be the corresponding cluster algebra. Let $H$ be the path algebra of $Q$ over an algebraically closed field and let $M$ be an indecomposable…

Euler characteristic of quiver Grassmannians and Ringel-Hall algebras of string algebras

- Mathematics
- 2010

We compute the Euler characteristics of quiver Grassmannians and quiver flag varieties of tree and band modules and prove their positivity. This generalizes some results by G.C. Irelli…

Quiver Grassmannians associated with string modules

- Mathematics
- 2011

We provide a technique to compute the Euler---Poincare characteristic of a class of projective varieties called quiver Grassmannians. This technique applies to quiver Grassmannians associated with…

Cluster Multiplication in Regular Components via Generalized Chebyshev Polynomials

- Mathematics, Computer Science
- 2008

It is proved that these polynomials arise in the context of cluster characters associated to Dynkin quivers of type A and representation-infinite quivers to obtain a simple combinatorial description of cluster algebras of type $ \mathbb{A} $.

On triangulated orbit categories

- Mathematics
- 2005

We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by A. Buan, R.…

Quiver varieties and cluster algebras

- Mathematics
- 2011

Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of…

Cluster algebras II: Finite type classification

- Mathematics
- 2002

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many…