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2-frieze patterns and the cluster structure of the space of polygons

Sophie Morier-Genoud, Valentin Ovsienko, Serge Tabachnikov (2012)

Annales de l’institut Fourier

We study 2-frieze patterns generalizing that of the classical Coxeter-Conway frieze patterns. The geometric realization of this space is the space of n -gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus 0 curves with n marked points. We show that the space of 2-frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.

A cluster algebra approach to q -characters of Kirillov–Reshetikhin modules

David Hernandez, Bernard Leclerc (2016)

Journal of the European Mathematical Society

We describe a cluster algebra algorithm for calculating q -characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra U q ( 𝔤 ^ ) . This yields a geometric q -character formula for tensor products of Kirillov–Reshetikhin modules. When 𝔤 is of type A , D , E , this formula extends Nakajima’s formula for q -characters of standard modules in terms of homology of graded quiver varieties.

Bar-invariant bases of the quantum cluster algebra of type A 2 ( 2 )

Xueqing Chen, Ming Ding, Jie Sheng (2011)

Czechoslovak Mathematical Journal

We construct bar-invariant [ q ± 1 / 2 ] -bases of the quantum cluster algebra of the valued quiver A 2 ( 2 ) , one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.

Cambrian fans

Nathan Reading, David E. Speyer (2009)

Journal of the European Mathematical Society

For a finite Coxeter group W and a Coxeter element c of W ; the c -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W . Its maximal cones are naturally indexed by the c -sortable elements of W . The main result of this paper is that the known bijection cl c between c -sortable elements and c -clusters induces a combinatorial isomorphism of fans. In particular, the c -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W . The rays...

Cluster categories for algebras of global dimension 2 and quivers with potential

Claire Amiot (2009)

Annales de l’institut Fourier

Let k be a field and A a finite-dimensional k -algebra of global dimension 2 . We construct a triangulated category 𝒞 A associated to A which, if  A is hereditary, is triangle equivalent to the cluster category of A . When 𝒞 A is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also...

Component clusters for acyclic quivers

Sarah Scherotzke (2016)

Colloquium Mathematicae

The theory of Caldero-Chapoton algebras of Cerulli Irelli, Labardini-Fragoso and Schröer (2015) leads to a refinement of the notions of cluster variables and clusters, via so-called component clusters. We compare component clusters to classical clusters for the cluster algebra of an acyclic quiver. We propose a definition of mutation between component clusters and determine the mutation relations of component clusters for affine quivers. In the case of a wild quiver, we provide bounds for the size...

Dimers and cluster integrable systems

Alexander B. Goncharov, Richard Kenyon (2013)

Annales scientifiques de l'École Normale Supérieure

We show that the dimer model on a bipartite graph Γ on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space Ł Γ of line bundles with connections on the graph Γ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs Γ 1 and Γ 2 areequivalentif the Newton polygons of the corresponding partition functions...

Finite mutation classes of coloured quivers

Hermund André Torkildsen (2011)

Colloquium Mathematicae

We show that the mutation class of a coloured quiver arising from an m-cluster tilting object associated with a finite-dimensional hereditary algebra H, is finite if and only if H is of finite or tame representation type, or it has at most two simples. This generalizes a result known for cluster categories.

Higher-dimensional cluster combinatorics and representation theory

Steffen Oppermann, Hugh Thomas (2012)

Journal of the European Mathematical Society

Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite-dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection...

Mutating seeds: types 𝔸 and 𝔸 ˜ .

Ibrahim Assem, Christophe Reutenauer (2012)

Annales mathématiques Blaise Pascal

In the cases 𝔸 and 𝔸 ˜ , we describe the seeds obtained by sequences of mutations from an initial seed. In the 𝔸 ˜ case, we deduce a linear representation of the group of mutations which contains as matrix entries all cluster variables obtained after an arbitrary sequence of mutations (this sequence is an element of the group). Nontransjective variables correspond to certain subgroups of finite index. A noncommutative rational series is constructed, which contains all this information.

Skew-symmetric cluster algebras of finite mutation type

Anna Felikson, Michael Shapiro, Pavel Tumarkin (2012)

Journal of the European Mathematical Society

In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices....

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