## 38 Citations

LINEAR RELATIONS AND INTEGRABILITY FOR CLUSTER ALGEBRAS FROM AFFINE QUIVERS

- MathematicsGlasgow Mathematical Journal
- 2020

Abstract We consider frieze sequences corresponding to sequences of cluster mutations for affine D- and E-type quivers. We show that the cluster variables satisfy linear recurrences with periodic…

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…

Linear recurrence relations in Q-systems and difference L-operators

- Mathematics
- 2014

We study linear recurrence relations in the character solutions of Q-systems obtained from the Kirillov–Reshetikhin modules. We explain how known results on difference L-operators lead to a uniform…

Representation type via Euler characteristics and singularities of quiver Grassmannians

- MathematicsBulletin of the London Mathematical Society
- 2019

In this paper, we characterize the representation type of an acyclic quiver by the properties of its associated quiver Grassmannians. This characterization utilizes and extends known results about…

A family of linearizable recurrences with the Laurent property

- Mathematics
- 2014

We consider a family of non‐linear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of…

Cluster mutation-periodic quivers and associated Laurent sequences

- Mathematics
- 2011

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the…

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…

LINEAR RECURRENCE RELATIONS IN Q-SYSTEMS VIA LATTICE POINTS IN POLYHEDRA

- MathematicsTransformation Groups
- 2018

We prove that the sequence of the characters of the Kirillov–Reshetikhin (KR) modules Wma$$ {W}_m^{(a)} $$, m ∈ ℤm≥0 associated to a node a of the Dynkin diagram of a complex simple Lie algebra g$$…

Quiver Grassmannians of extended Dynkin type D - Part 2: Schubert decompositions and F-polynomials

- Mathematics
- 2015

Extending the main result of Part 1, in the first part of this paper we show that every quiver Grassmannian of a representation of a quiver of extended Dynkin type $D$ has a decomposition into affine…

## References

SHOWING 1-10 OF 38 REFERENCES

Cluster algebras as Hall algebras of quiver representations

- Mathematics
- 2004

Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be…

Generalized associahedra via quiver representations

- Mathematics
- 2002

We provide a quiver-theoretic interpretation of certain smooth complete simplicial fans associated to arbitrary finite root systems in recent work of S. Fomin and A. Zelevinsky. The main properties…

Cluster mutation-periodic quivers and associated Laurent sequences

- Mathematics
- 2011

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the…

Cluster algebras I: Foundations

- Mathematics
- 2001

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

Triangulated Categories: Cluster algebras, quiver representations and triangulated categories

- Mathematics
- 2010

This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on…

Q-system Cluster Algebras, Paths and Total Positivity ?

- Mathematics
- 2010

In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show…

From triangulated categories to cluster algebras

- Mathematics
- 2005

The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra $\mathcal{A}$ of finite type can be realized as a…

Generic Variables in Acyclic Cluster Algebras and Bases in Affine Cluster Algebras

- Mathematics
- 2008

Let $Q$ be a finite quiver without oriented cycles and $\mathcal A(Q)$ be the coefficient-free cluster algebra with initial seed $(Q,\textbf u)$. Using the Caldero-Chapoton map, we introduce and…