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Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

An exchange matrix B dominates an exchange matrix B' if the signs of corresponding entries weakly agree, with the entry of B always having weakly greater absolute value. When B dominates B', interesting things happen in many cases (but not always): the identity map between the associated mutation-linear structures is often mutation-linear; the mutation fan for B often refines the mutation fan for B'; the scattering (diagram) fan for B often refines the scattering fan for B'; and there is often an injective homomorphism from the principal-coefficients cluster algebra for B' to the principal-coefficients cluster algebra for B, preserving \mathbf {g}-vectors and sending the set of cluster variables for B' (or an analogous larger set) into the set of cluster variables for B (or an analogous larger set). The scope of the description "often" is not the same in all four contexts and is not settled in any of them. In this paper, we prove theorems that provide examples of these dominance phenomena. [ABSTRACT FROM AUTHOR]

Let C be a 2-Calabi-Yau triangulated category, and let T be a cluster tilting subcategory of C. An important result from Dehy and Keller tells us that a rigid object c ∈ C is uniquely defined by its index with respect to T. The notion of triangulated categories extends to the notion of (d + 2)-angulated categories. Thanks to a paper by Oppermann and Thomas, we now have a definition for cluster tilting subcategories in higher dimensions. This paper proves that under a technical assumption, an indecomposable object in a (d + 2)-angulated category is uniquely defined by its index with respect to a higher dimensional cluster tilting subcategory. We also demonstrate that this result applies to higher dimensional cluster categories of Dynkin-type A. [ABSTRACT FROM AUTHOR]

Let g be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism D on the coordinate ring C[N] related to Brion's equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram's strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when g is of type An or D4, the map D takes similar distinguished values on the set of all flag minors of C[N], raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of D on flag minors belonging to the same standard seed, and we show that in any ADE type these relations are preserved under cluster mutations from one standard seed to another. The proofs of these results partly rely on Kang-Kashiwara-Kim-Oh's monoidal categorification of the cluster structure of C[N] via representations of quiver Hecke algebras. [ABSTRACT FROM AUTHOR]

A cluster algebra is unistructural if the set of its cluster variables determines its clusters and seeds. It is conjectured that all cluster algebras are unistructural. In this paper, we show that any cluster algebra arising from a triangulation of a marked surface without punctures is unistructural. Our proof relies on the existence of a positive basis known as the bracelet basis and on the skein relations. We also prove that a cluster algebra defined from a disjoint union of quivers is unistructural if and only if the cluster algebras defined from the connected components of the quiver are unistructural. [ABSTRACT FROM AUTHOR]

In this paper, we describe a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well-studied case of dimer models on a disc. We prove that all Berenstein-Fomin-Zelevinsky quivers for Schubert cells in a symmetric Kac-Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten, and Smith that the corresponding superpotentials are rigid using the dimer model structure of the quivers. [ABSTRACT FROM AUTHOR]

A universal geometric cluster algebra over an exchange matrix B is a universal object in the category of geometric cluster algebras over B related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster algebras relative to the definition originally given by Fomin and Zelevinsky.) The universal objects are closely related to a fan FB called the mutation fan for B. In this paper, we consider universal geometric cluster algebras and mutation fans for cluster algebras arising from marked surfaces. We identify two crucial properties of marked surfaces: The Curve Separation Property and the Null Tangle Property. The latter property implies the former. We prove the Curve Separation Property for all marked surfaces except once-punctured surfaces without boundary components, and as a result we obtain a construction of the rational part of FB for these surfaces. We prove the Null Tangle Property for a smaller family of surfaces and use it to construct universal geometric coefficients for these surfaces. [ABSTRACT FROM AUTHOR]

There is a well-known analogy between cluster algebras and Kacmoody algebras: roughly speaking, Kac-Moody algebras are associated with symmetrizable generalized Cartan matrices while cluster algebras correspond to skew-symmetrizable matrices. In this paper, we study an interplay between these two classes of matrices. We obtain relations in the Weyl groups of Kac- Moody algebras that come from mutation classes of skew-symmetrizable matrices. More precisely, we establish a set of relations satisfied by the reflections of the so-called companion bases; these include c-vectors, which parametrize coefficients in a cluster algebra with principal coefficients. These relations generalize the relations obtained by Barot and Marsh for finite type. For affine type, we also show that the reflections of the companion bases satisfy the relations obtained by Felikson and Tumarkin. As an application, we obtain some combinatorial properties of the mutation classes of skew-symmetrizable matrices. [ABSTRACT FROM AUTHOR]

The sine-Gordon Y-systems and the reduced sine-Gordon Y- systems were introduced by Tateo in the 1990's in the study of the integrable deformation of conformal field theory by the thermodynamic Bethe ansatz method. The periodicity property and the dilogarithm identities concerning these Y-systems were conjectured by Tateo, and only a part of them have been proved so far. In this paper we formulate these Y-systems by the polygon realization of cluster algebras of types A and D and prove the conjectured periodicity and dilogarithm identities in full generality. As it turns out, there is a wonderful interplay among continued fractions, triangulations of polygons, cluster algebras, and Y-systems. [ABSTRACT FROM AUTHOR]

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$ $ \pi_\downarrow^c$ to the unique maximal $ c$ $ \pi_\downarrow^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 $ \pi_\downarrow^c$ [ABSTRACT FROM AUTHOR]

In the structural theory of cluster algebras, a crucial role is played by a family of integer vectors, called c-vectors, which parametrize the coefficients. It has recently been shown that each c-vector with respect to an acyclic initial seed is a real root of the corresponding root system. In this paper, we obtain an interpretation of this result in terms of symmetric matrices. We show that for skew-symmetric cluster algebras, the c-vectors associated with any seed defines a quasi-Cartan companion for the corresponding exchange matrix (i.e. they form a companion basis), and we establish some basic combinatorial properties. In particular, we show that these vectors define an admissible cut of edges in the associated quivers. [ABSTRACT FROM AUTHOR]

Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced by Fomin and Zelevinsky in 2000 in the context of Lie theory, but have since appeared in many other contexts, from Poisson geometry to triangulations of surfaces and Teichmüller theory. In this expository paper we give an introduction to cluster algebras, and illustrate how this framework naturally arises in Teichmüller theory. We then sketch how the theory of cluster algebras led to a proof of the Zamolodchikov periodicity conjecture in mathematical physics. [ABSTRACT FROM AUTHOR]

The article reports on a paper that discusses the prominent role played, in the recent literature on cluster algebras, by calculations of Euler characteristics of certain types associated to quiver representations. The Chamber Ansatz of Arkady Berenstein, Sergei Fomin, and Andrei Zelevinsky, the very source of cluster algebras, are referred to. Topics discussed include the generic bases for cluster algebras, the categorifications of the twist automorphism and of the Chamber Ansatz.

There is a particular analogy between combinatorial aspects of cluster algebras and Kac-Moody algebras: roughly speaking, cluster algebras are associated with skew-symmetrizable matrices while Kac-Moody algebras correspond to (symmetrizable) generalized Cartan matrices. Both classes of algebras and the associated matrices have the same classification of finite type objects by the well-known Cartan-Killing types. In this paper, we study an extension of this correspondence to the affine type. In particular, we establish the cluster algebras which are determined by the generalized Cartan matrices of affine type. [ABSTRACT FROM AUTHOR]