Cluster Algebras Of Type Research Articles

A categorification of cluster algebras of type B and C through symmetric quivers

We express cluster variables of type Bn and Cn in terms of cluster variables of type An. Then we associate a cluster tilted bound symmetric quiver Q of type A2n−1 to any seed of a cluster algebra of type Bn and Cn. Under this correspondence, cluster variables of type Bn (resp. Cn) correspond to orthogonal (resp. symplectic) indecomposable representations of Q. We find a Caldero-Chapoton map in this setting. We also give a categorical interpretation of the cluster expansion formula in the case of acyclic quivers.

The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces

The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. The precise relations between the formulas counting faces and the one counting chains of noncrossing partitions are combinatorial reciprocities, generalizing the one between Narayana and Kirkman numbers.

Deformations of cluster mutations and invariant presymplectic forms

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4, we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions in the discrete sine-Gordon equation.

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

By viewing $$\tilde{A}$$ and $$\tilde{D}$$ 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 previously found for the cluster map associated with these frieze patterns. We show that these cluster variables form friezes which are precisely the ones found by Assem–Dupont by applying the cluster character to the associated cluster category.

An Expansion Formula for Decorated Super-Teichmüller Spaces

Motivated by the definition of super-Teichm\"uller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichm\"uller space, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super $\lambda$-lengths associated to arcs in a bordered surface. In the special case of a disk, we are able to give combinatorial expansion formulas for the super $\lambda$-lengths associated to diagonals of a polygon in the spirit of Ralf Schiffler's $T$-path formulas for type $A$ cluster algebras. We further connect our formulas to the super-friezes of Morier-Genoud, Ovsienko, and Tabachnikov, and obtain partial progress towards defining super cluster algebras of type $A_n$. In particular, following Penner-Zeitlin, we are able to get formulas (up to signs) for the $\mu$-invariants associated to triangles in a triangulated polygon, and explain how these provide a step towards understanding odd variables of a super cluster algebra.

CLUSTER ALGEBRAS FROM SURFACES AND EXTENDED AFFINE WEYL GROUPS

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangulation a basis in V , such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type A, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types E.

Cluster algebras arising from cluster tubes I: integer vectors

We study cluster algebras arising from cluster tubes. We obtain categorical interpretations for $g$-vectors, $c$-vectors and denominator vectors for cluster algebras of type $\mathrm{C}$ with respect to arbitrary initial seeds. In particular, a denominator theorem has been proved, which enables us to establish the linearly independence of denominator vectors of cluster variables from the same cluster for cluster algebras of type $\mathrm{A}\mathrm{B}\mathrm{C}$. This strengthens the link between cluster tubes and cluster algebras of type $\mathrm{C}$ initiated by Buan, Marsh and Vatne.

Heronian Friezes

Abstract Motivated by computational geometry of point configurations on the Euclidean plane, and by the theory of cluster algebras of type $A$, we introduce and study Heronian friezes, the Euclidean analogues of Coxeter’s frieze patterns. We prove that a generic Heronian frieze possesses the glide symmetry (hence is periodic) and establish the appropriate version of the Laurent phenomenon. For a closely related family of Cayley–Menger friezes, we identify an algebraic condition of coherence, which all friezes of geometric origin satisfy. This yields an unambiguous propagation rule for coherent Cayley–Menger friezes, as well as the corresponding periodicity results.

Tensor products and q-characters of HL-modules and monoidal categorifications

We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite--dimensional representations of a quantum affine algebra of type $A$. We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type $A$ with coefficients.

Cluster automorphisms and quasi-automorphisms

We study the relation between the cluster automorphisms and the quasi-automorphisms of a cluster algebra A. We prove that under some mild condition, satisfied for example by every skew-symmetric cluster algebra, the quasi-automorphism group of A is isomorphic to a subgroup of the cluster automorphism group of Atriv, and the two groups are isomorphic if A has principal or universal coefficients; here Atriv is the cluster algebra with trivial coefficients obtained from A by setting all frozen variables equal to the integer 1.We also compute the quasi-automorphism group of all finite type and all skew-symmetric affine type cluster algebras, and show in which types it is isomorphic to the cluster automorphism group of Atriv.

Generators of rank 2 cluster algebras of affine types via linearization of seed mutations

From the viewpoint of integrable systems on algebraic curves, we discuss linearization of birational maps arising from the seed mutations of types A1(1) and A2(2), which enables us to construct the set of all cluster variables generating the corresponding cluster algebras. These birational maps induce discrete integrable systems on algebraic curves referred to as the types of the seed mutations from which they are arising. The invariant curve of type A1(1) is a conic, while the one of type A2(2) is a singular quartic curve. By applying the blowing-up of the singular quartic curve, the discrete integrable system of type A2(2) on the singular curve is transformed into the one on the conic, the invariant curve of type A1(1). We show that both the discrete integrable systems of types A1(1) and A2(2) commute with each other on the conic, the common invariant curve. We moreover show that these integrable systems are simultaneously linearized by means of the conserved quantities and their general solutions are obtained. By using the general solutions, we construct the sets of all cluster variables generating the cluster algebras of types A1(1) and A2(2).

Cluster multiplication theorem in the quantum cluster algebra of type [formula omitted] and the triangular basis

The objective of the present paper is to prove cluster multiplication theorem in the quantum cluster algebra of type A2(2). As corollaries, we obtain bar-invariant Z[q±12]-bases established in [6], and naturally deduce the positivity of the elements in these bases. One bar-invariant basis as the triangular basis of this quantum cluster algebra is also explicitly described.

Cluster algebraic interpretation of infinite friezes

Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to matching numbers in the once-punctured disk and annulus. In this paper, we study such infinite friezes with an eye towards cluster algebras of type D and affine A, respectively. By examining infinite friezes with Laurent polynomial entries, we discover new symmetries and formulas relating the entries of this frieze to one another. Lastly, we also present a correspondence between Broline, Crowe and Isaacs’s classical matching tuples and combinatorial interpretations of elements of cluster algebras from surfaces.

Combinatorics of X-variables in finite type cluster algebras

We compute the number of X-variables (also called coefficients) of a cluster algebra of finite type when the underlying semifield is the universal semifield. For classical types, these numbers arise from a bijection between coefficients and quadrilaterals (with a choice of diagonal) appearing in triangulations of certain marked surfaces. We conjecture that similar results hold for cluster algebras from arbitrary marked surfaces, and obtain corollaries regarding the structure of finite type cluster algebras of geometric type.

Friezes and continuant polynomials with parameters

Frieze patterns (in the sense of Conway and Coxeter) are related to cluster algebras of type A and to signed continuant polynomials. In view of studying certain classes of cluster algebras with coefficients, we extend the concept of signed continuant polynomial to define a new family of friezes, called c-friezes, which generalises frieze patterns. Having in mind the cluster algebras of finite type, we identify a necessary and sufficient condition for obtaining periodic c-friezes. Taking into account the Laurent phenomenon and the positivity conjecture, we present ways of generating c-friezes of integers and of positive integers. We also show some specific properties of c-friezes.

Applications of graded methods to cluster variables in arbitrary types

This thesis is concerned with studying the properties of gradings on several examples of cluster algebras, primarily of infinite type. We start by considering two classes of finite type cluster algebras: those of type Bn and Cn. We give the number of cluster variables of each occurring degree and verify that the grading is balanced. These results complete a classification in [16] for coefficient-free finite type cluster algebras. We then consider gradings on cluster algebras generated by 3×3 skew-symmetric matrices. We show that the mutation-cyclic matrices give rise to gradings in which all occurring degrees are positive and have only finitely many associated cluster variables (excepting one particular case). For the mutation-acyclic matrices, we prove that all occurring degrees have infinitely many variables and give a direct proof that the gradings are balanced. We provide a condition for a graded cluster algebra generated by a quiver to have infinitely many degrees, based on the presence of a subquiver in its mutation class. We use this to study the gradings on cluster algebras that are (quantum) coordinate rings of matrices and Grassmannians and show that they contain cluster variables of all degrees in N. Next we consider the finite list (given in [9]) of mutation-finite quivers that do not correspond to triangulations of marked surfaces. We show that A(X7) has a grading in which there are only two degrees, with infinitely many cluster variables in both. We also show that the gradings arising from Ee6, Ee7 and Ee8 have infinitely many variables in certain degrees. Finally, we study gradings arising from triangulations of marked bordered 2- dimensional surfaces (see [10]). We adapt a definition from [24] to define the space of valuation functions on such a surface and prove combinatorially that this space is isomorphic to the space of gradings on the associated cluster algebra. We illustrate this theory by applying it to a family of examples, namely, the annulus with n + m marked points. We show that the standard grading is of mixed type, with finitely many variables in some degrees and infinitely many in the others. We also give an alternative grading in which all degrees have infinitely many cluster variables.

Towards a uniform subword complex description of acyclic finite type cluster algebras

It has been established in recent years how to approach acyclic cluster algebras of finite type using subword complexes. We continue this study by uniformly describing the c- and g-vectors, and by providing a conjectured description of the Newton polytopes of the F-polynomials. We moreover show that this conjectured description would imply that finite type cluster complexes are realized by the duals of the Minkowski sums of the Newton polytopes of either the F-polynomials or of the cluster variables, respectively. We prove this conjectured description to hold in type A and in all types of rank at most 8 including all exceptional types, leaving types B, C, and D conjectural.

Cluster Expansion Formulas in Type A

The aim of this paper is to give analogs of the cluster expansion formula of Musiker and Schiffler for cluster algebras of type A with coefficients arising from boundary arcs of the corresponding triangulated polygon. Indeed, we give three cluster expansion formulas by perfect matchings of angles in triangulated polygon, by discrete subsets of arrows of the corresponding ice quiver and by minimal cuts of the corresponding quiver with potential.

Bases for cluster algebras from orbifolds

We generalize the construction of the bracelet and bangle bases defined in [36] and the band basis defined in [43] to cluster algebras arising from orbifolds. We prove that the bracelet bases are positive, and the bracelet basis for the affine cluster algebra of type Cn(1) is atomic. We also show that cluster monomial bases of all skew-symmetrizable cluster algebras of finite type are atomic.

Artin Group Presentations Arising from Cluster Algebras

In 2003, Fomin and Zelevinsky proved that finite type cluster algebras can be classified by Dynkin diagrams. Then in 2013, Barot and Marsh defined the presentation of a reflection group associated to a Dynkin diagram in terms of an edge-weighted, oriented graph, and proved that this group is invariant (up to isomorphism) under diagram mutations. In this paper, we extend Barot and Marsh’s results to Artin group presentations, defining new generator relations and showing mutation-invariance for these presentations.