Cluster Algebras Of Finite Type Research Articles

C-vectors via τ-tilting theory

Inspired by the tropical dualities in cluster algebras, we introduce c-vectors for finite-dimensional algebras via τ-tilting theory. Let A be a finite-dimensional algebra over an algebraically closed field k. Each c-vector of A can be realized as the (negative) dimension vector of certain indecomposable A-module and hence we establish the sign-coherence property for this kind of c-vectors. We then study the positive c-vectors for certain classes of finite-dimensional algebras including quasitilted algebras and cluster-tilted algebras. In particular, we recover the equalities of c-vectors for acyclic cluster algebras and skew-symmetric cluster algebras of finite type respectively obtained by Nájera Chávez. To this end, a short proof for the sign-coherence of c-vectors for skew-symmetric cluster algebras has been given in the appendix.

Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients

We give an explicit description of all the exchange relations in any finite type cluster algebra with acyclic initial seed and principal coefficients.

Polynomial recognition of cluster algebras of finite type

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. Using the standard definition, the problem is infeasible since it uses mutations that can lead to an infinite process. Barot, Geiss and Zelevinsky [1] presented an easier way to verify if a given algebra is of finite type, by testing if all chordless cycles of the graph related to the algebra are cyclically oriented and if there exists a positive quasi-Cartan companion of the skew-symmetrizable matrix related to the algebra. We develop an algorithm that verifies these conditions and decides whether or not a cluster algebra is of finite type in polynomial time. The second part of the algorithm is used to prove that the more general problem to decide if a matrix has a positive quasi-Cartan companion is in NP class.

Cluster automorphism groups of cluster algebras of finite type

We study the cluster automorphism group Aut(A) of a coefficient free cluster algebra A of finite type. A cluster automorphism of A is a permutation of the cluster variable set X that is compatible with cluster mutations. We show that, on the one hand, by the well-known correspondence between X and the almost positive root system Φ≥−1 of the corresponding Dynkin type, the piecewise-linear transformations τ+ and τ− on Φ≥−1 induce cluster automorphisms f+ and f− of A respectively; on the other hand, excepting type D2n (n⩾2), all the cluster automorphisms of A are compositions of f+ and f−. For a cluster algebra of type D2n (n⩾2), there exists an exceptional cluster automorphism induced by a permutation of negative simple roots in Φ≥−1, which is not a composition of τ+ and τ−. By using these results and folding a simply laced cluster algebra, we compute the cluster automorphism group for a non-simply laced finite type cluster algebra. As an application, we show that Aut(A) is isomorphic to the cluster automorphism group of the FZ-universal cluster algebra of A.

Graded cluster algebras

In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite-type cluster algebras without coefficients, giving a full classification. Translating the definition suitably again, we obtain a notion of multi-grading for (generalised) cluster categories. This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases. We also obtain interesting combinatorics--namely tropical frieze patterns--on the Auslander---Reiten quivers of the categories.

The diameter of type [formula omitted] associahedra and the non-leaving-face property

Generalized associahedra were introduced by S. Fomin and A. Zelevinsky in connection to finite type cluster algebras. Following recent work of L. Pournin in types A and B, this paper focuses on geodesic properties of generalized associahedra. We prove that the graph diameter of the n-dimensional associahedron of type D is precisely 2n−2 for all n greater than 1. Furthermore, we show that all type BCD associahedra have the non-leaving-face property, that is, any geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both. This property was already proven by D. Sleator, R. Tarjan and W. Thurston for associahedra of type A. In contrast, we present relevant examples related to the associahedron that do not always satisfy this property.

Reflection group presentations arising from cluster algebras

We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.

Denominator vectors and compatibility degrees in cluster algebras of finite type

We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra. They provide two simple proofs of the known fact that the d d -vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero.

Diagrammatic Description of $c$-Vectors and $d$-Vectors of Cluster Algebras of Finite Type

We provide an explicit Dynkin diagrammatic description of the $c$-vectors and the $d$-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface realization of cluster algebras for types $A_n$ and $D_n$, then we apply the folding method to $D_{n+1}$ and $A_{2n-1}$ to obtain types $B_n$ and $C_n$. Exceptional types are done by direct inspection with the help of a computer algebra software. We also propose a conjecture on the root property of $c$-vectors for a general cluster algebra.

Mutation classes of finite type cluster algebras with principal coefficients

Cluster algebras of finite type is a fundamental class of algebras whose classification is identical to the famous Cartan–Killing classification. More recently, Fomin and Zelevinsky introduced another central notion of cluster algebras with principal coefficients. These algebras are determined combinatorially by mutation classes of certain rectangular matrices. It was conjectured, by Fomin and Zelevinsky, that finite type cluster algebras with principal coefficients are characterized by the mutation classes which are finite. In this paper, we prove this conjecture.

Denominator vectors and compatibility degrees in cluster algebras of finite type

We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra, and provide simple proofs of the known fact that the $d$-vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero. Nous présentons deux descriptions élémentaires des vecteurs dénominateurs des algèbres amassées de type fini pour tout amas initial: l'une en termes de degrés de compatibilitié entre racines presque positives définis par S. Fomin et A. Zelevinsky, et l'autre en termes de la fonction racine d'un certain complexe de sous-mots. Ces descriptions ne reposent que sur l'algèbre linéaire et fournissent des preuves simples du fait (connu) que le $d$-vecteur de toute variable d'amas, qui n'est pas dans l'amas initial, a des entrées positives ou nulles et est différent du vecteur nul.

Skew-symmetric cluster algebras of finite mutation type

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. Besides cluster algebras of rank 2 and cluster algebras associated with triangulations of surfaces there are exactly 11 exceptional skew-symmetric cluster algebras of finite mutation type. More precisely, 9 of them are associated with root systems E_6 , E_7 , E_8 , \widetilde E_6 , \widetilde E_7 , \widetilde E_8 , E_6^{(1,1)} , E_7^{(1,1)} , E_8^{(1,1)} ; two remaining were found by Derksen and Owen in [DO]. We also describe a criterion which determines if a skew-symmetric cluster algebra is of finite mutation type, and discuss growth rate of cluster algebras.

A Graph Theoretic Expansion Formula for Cluster Algebras of Classical Type

In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type with bipartite seed. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the A n case while providing a novel interpretation for the B n , C n , and D n cases.

On quantum cluster algebras of finite type

We extend the definition of a quantum analogue of the Caldero-Chapoton map defined by D. Rupel. When Q is a quiver of finite type, we prove that the algebra (Q) generated by all cluster characters is exactly the quantum cluster algebra (Q).

On a Quantum Analog of the Caldero-Chapoton Formula

Let $Q$ be any invertible valued quiver without oriented cycles. We study connections between the category of valued representations of $Q$ and expansions of cluster variables in terms of the initial cluster in quantum cluster algebras. We show that an analogue of the Caldero-Chapoton formula holds for all quantum cluster algebras of finite type and for any cluster variable in an almost acyclic cluster.

Sortable elements in infinite Coxeter groups

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection \pidown^c mapping each element of W to the unique maximal c-sortable element below it in the weak order. The fibers of \pidown^c essentially define the c-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of \pidown^c. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.

Cyclic sieving and cluster multicomplexes

Reiner, Stanton and White (2004) [10] proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu (2008) [2] generalized these results to Cartan–Killing types other than A by means of actions of deformed Coxeter elements on cluster complexes of Fomin and Zelevinsky (2003) [6]. The Reiner–Stanton–White and Eu–Fu results were proven using direct counting arguments. We give representation theoretic proofs of closely related results using the notion of noncrossing and seminoncrossing tableaux due to Pylyavskyy (2009) [9] as well as some geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky (2003) [5].

Cluster Algebras of Finite Type via Coxeter Elements and Principal Minors

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 the simply connected semisimple algebraic group of the same Cartan–Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors.

A Cluster Expansion Formula ($A_n$ case)

We consider the Ptolemy cluster algebras, which are cluster algebras of finite type $A$ (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed $\Sigma$ in a Ptolemy cluster algebra, we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed $\Sigma$. Our formula is given in a combinatorial way, using paths on a triangulation of the polygon that corresponds to the seed $\Sigma$.

An approach to non-simply laced cluster algebras

We realize cluster algebras of non-simply laced types as subalgebras of a certain quotient of cluster algebras associated to quivers with automorphisms. We prove that equality holds for finite type cluster algebras. We also give an interpretation of this results in terms of invariant objects under the action of the automorphism group on the corresponding cluster category.