Figure 5 - uploaded by Vincent Pilaud
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Three possible flips in types B and C: either the quadrilateral involves no long diagonal (left), or one long diagonal is an edge of the quadrilateral (middle), or the two diagonals of the quadrilateral are long diagonals (right).
Source publication
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....
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Citations
... Graph isomorphismθ c . The graph isomorphism between the quiver of c-clusters and a certain exchange quiver is provided in [CP15]. We fix a Dynkin-type root system Φ and a Coxeter element c ∈ W (Φ). We obtain a skew-symmetrizable matrix B c = (b ij ) from the Cartan matrix C Φ = (C ij ) and c as follows: ...
... Theorem 3.1 ( [CP15]). We fix a Dynkin-type root system Φ of rank n and a Coxeter element c ∈ W (Φ). Let X (B c ) represent the set of cluster variables in A(B c ; t 0 ). ...
... is initial, and this conflicts with Proposition 5.5. Now, we recall the interpretation of τ c by mutations provided in [CP15]. ...
In this paper, we demonstrate that the lattice structure of a set of clusters in a cluster algebra of finite type is anti-isomorphic to the torsion lattice of a certain Geiss-Leclerc-Schr\"oer (GLS) path algebra and to the $c$-Cambrian lattice. We prove this by explicitly describing the exchange quivers of cluster algebras of finite type. Specifically, we prove that these quivers are anti-isomorphic to those formed by support $\tau$-tilting modules in GLS path algebras and to those formed by $c$-clusters consisting of almost positive roots.
... Compatibility degree [33] is used to verify the rationality of task assignment, which can be expressed as: ...
Mobile crowdsensing (MCS) offers a novel paradigm for large-scale sensing with the proliferation of smartphones. Task assignment is a critical problem in mobile crowdsensing (MCS), where service providers attempt to recruit a group of brilliant users to complete the sensing task at a limited cost. However, selecting an appropriate set of users with high quality and low cost is challenging. Existing works of task assignment ignore the data redundancy of large-scale users and the individual preference of service providers, resulting in a significant workload on the sensing platform and inaccurate assignment results. To tackle this issue, we propose a task assignment method based on user-union clustering and individual preferences, which considers the influence of clustering data quality and preference-based sensing cost. Firstly, we design a user-union clustering algorithm (UCA) by defining user similarity and setting user scale, which aims to balance user distribution, reduce data redundancy, and improve the accuracy of high-quality user aggregation. Then, we consider individual preferences of service providers and construct a preference-based task assignment algorithm (PTA) to achieve the diversified sensing cost control needs. To evaluate the performance of the proposed solutions, extensive simulations are conducted. The results demonstrate that our proposed solutions outperform the baseline algorithm, which realizes the individual preference-based task assignment under the premise of ensuring high-quality data.
... We remark that the statement (1) of Theorem 4.5 has been known for any cluster algebras of finite type (cf. [12,15]) and has been recently verified for skew-symmetric type in [13]. We also remark here that this conjecture has been proved for any skew-symmetrizable cluster algebra by using different methods in the arXiv paper [14] after the current paper was posted on the arXiv. ...
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.
... For cluster algebras of finite type, Conjecture 1 is proved in [6]. For cluster algebras from surfaces, the components of d-vectors are interpreted as certain modified intersection numbers (see [11]). ...
... The compatibility degree on the set of cluster variables was firstly defined in [6] for cluster algebras of finite type. It needs Conjecture 1 (ii), which was proved for cluster algebras of finite type in [6], to guarantee the validity of the definition of compatibility degree. ...
... The compatibility degree on the set of cluster variables was firstly defined in [6] for cluster algebras of finite type. It needs Conjecture 1 (ii), which was proved for cluster algebras of finite type in [6], to guarantee the validity of the definition of compatibility degree. Thanks to Theorem 11, we can extend the definition of compatibility degree to any skew-symmetrizable cluster algebras. ...
In this paper, we introduce the enough $g$-pairs property for principal coefficients cluster algebras, which can be understood as a strong version of the sign-coherence of the $G$-matrices. Then we prove that any skew-symmetrizable principal coefficients cluster algebra has the enough $g$-pairs property.
As applications, we prove some long standing conjectures in cluster algebras, including a conjecture on denominator vectors and a conjecture on exchange graphs (see Conjecture \ref{conjecture} and Conjecture \ref{conjecture2} below). In addition, we give a criterion to distinguish whether particular cluster variables belong to one common cluster for any skew-symmetrizable cluster algebra. As a corollary, we prove \cite[Conjecture 5.5]{FST} by Fomin, Shapiro, and Thurston.
... in the denominator of x. In finite type, this value was interpreted in [FZ03,CP15] can be used to construct a simplicial fan realization of the cluster complex, called d-vector fan: for certain initial clusters in finite type cluster algebras, the cones generated by the d-vectors of all collections of compatible cluster variables form a complete simplicial fan realizing the cluster complex. S. Fomin and A. Zelevinsky [FZ03] show it for the bipartite initial cluster, S. Stella [Ste13] for all acyclic initial clusters, and F. Santos [CSZ15, Section 5] for any initial cluster in type A. We expect this property to hold for any initial cluster of any finite type cluster algebra. ...
... We now show that our compatibility degree in path and cycle nested complexes matches the compatibility degree in the type A, B and C cluster complexes of S. Fomin and A. Zelevinsky [FZ03,CP15]. ...
International audience
Graph associahedra are polytopes realizing the nested complex N(G) on connected subgraphs of a graph G.While all known explicit constructions produce polytopes with the same normal fan, the great variety of fan realizationsof classical associahedra and the analogy between finite type cluster complexes and nested complexes incitedus to transpose S. Fomin and A. Zelevinsky's construction of compatibility fans for generalized associahedra (2003)to graph associahedra. Using a compatibility degree, we construct one fan realization of N(G) for each of its facets.Specifying G to paths and cycles, we recover a construction by F. Santos for classical associahedra (2011) and extendF. Chapoton, S. Fomin and A. Zelevinsky's construction (2002) for type B and C generalized associahedra.
... These results were generalized to arbitrary finite Coxeter groups by Ceballos, Labbé and Stump in [4]. The results in [4] provide an additional connection with the c-cluster complexes in the theory of cluster algebras, which has been used as a keystone for explicit results about denominator vectors in cluster algebras of finite type [5]. A construction of certain brick polytopes of spherical subword complexes is presented in [18,19], which is used to give a precise description of the toric varieties of c-generalized associahedra in connection with Bott-Samelson varieties in [7]. ...
... This function was introduced by Ceballos, Labbé and Stump in [4]. It encodes exchanges in the facets of the subword complex [4] and has been extensively used in the construction of Coxeter brick polytopes [19] and in the description of denominator vectors in cluster algebras of finite type [5]. ...
... The c-clusters are the sets of almost positive roots corresponding to clusters obtained by mutations from A c . These were described in purely combinatorial terms using a notion of c-compatibility relation by Reading in [20], and can be described purely in terms of the combinatorial models in the classical types [5,Section 7]. ...
International audience
We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras.
... Under this bijection, the (classical) compatibility degree of A(Φ) introduced in [FZ03b] is a function (· ·) cl from pairs of almost positive roots to nonnegative integers. The classical compatibility degree (· ·) cl has been further developed by Ceballos and Pilaud [CP15] to study denominator vectors for cluster algebras of finite type. Recently, Cao and Li [CL18] introduced a compatibility degree (we call it the dcompatibility degree) for any cluster complexes by using denominator vectors. ...
... We call it the classical compatiblity degree. In [FZ03b], the classical compatibility degree is called simply the compatibility degree, but we adopt this name in imitation of [CP15] to distinguish between it and forthcoming other degrees. For α, β ∈ Φ ≥−1 , we say that α and β are compatible if (α β) cl = (β α) cl = 0. ...
We introduce a new function on the set of pairs of cluster variables via $f$-vectors, which we call it the compatibility degree (of cluster complexes). The compatibility degree is a natural generalization of the classical compatibility degree introduced by Fomin and Zelevinsky. In particular, we prove that the compatibility degree has the duality property, the symmetry property, the embedding property and the compatibility property, which the classical one has. We also conjecture that the compatibility degree has the exchangeability property. As pieces of evidence of this conjecture, we establish the exchangeability property for cluster algebras of rank 2, acyclic skew-symmetric cluster algebras, cluster algebras arising from weighted projective lines, and cluster algebras arising from marked surfaces except for the once-punctured closed surfaces.
... It would be interesting to give a geometric or categorial explanation for compatibility degree function in general. Note that for cluster algebras of finite type or from surfaces, the compatibility degree function has nice explanation and one can refer to [8,9,5,12,6]. ...
We prove that any skew-symmetrizable cluster algebra is unistructural, which is a conjecture by Assem, Schiffler, and Shramchenko. As a corollary, we obtain that a cluster automorphism of a cluster algebra $\mathcal A(\mathcal S)$ is just an automorphism of the ambient field $\mathcal F$ which restricts to a permutation of cluster variables of $\mathcal A(S)$.
... The relation between the Tamari lattice and subword complexes has inspired further connections with pseudotriangulation polytopes [36,37], cluster algebras [12,19,20], Hopf algebras [6], and multiassociahedra [22,41]. Concerning the latter, the definition of the multiassociahedron can be naturally generalized to ν-trees, giving rise to the (k, ν)-Tamari complex, which is also a subword complex (these are the complexes of k-north-east fillings considered in [40]). ...
We give a new interpretation of the $\nu$-Tamari lattice of Pr\'eville-Ratelle and Viennot in terms of a rotation lattice of $\nu$-trees. This uncovers the relation with known combinatorial objects such as north-east fillings, tree-like tableaux and subword complexes. We provide a simple description of the lattice property using certain bracket vectors of $\nu$-trees, and show that the Hasse diagram of the $\nu$-Tamari lattice can be obtained as the facet adjacency graph of certain subword complex. Finally, this point of view generalizes to multi $\nu$-Tamari complexes, and gives (conjectural) insight on their geometric realizability via polytopal subdivisions of multiassociahedra.
... We remark that the statement (1) has been known for any cluster algebras of finite type (cf. [12,15]) and has been recently verified for skew-symmetric type in [14]. ...
We study cluster algebras of type $\mathrm{C}$ via representation theory of 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 an arbitrary initial seed. In particular, a denominator theorem has been proved. By applying the denominator theorem, we confirm certain conjectures on denominator vectors for cluster algebras of type $\mathrm{C}$. Among others, we establish the linearly independence of denominator vectors of cluster variables from the same cluster for cluster algebras of type $\mathrm{A}, \mathrm{B}$ or $\mathrm{C}$. Moreover, a Caldero-Chapoton formula has also been established for cluster algebras of type $\mathrm{C}$ with respect to arbitrary initial seeds. This strengthens the link between cluster tubes and cluster algebras of type $\mathrm{C}$ initialed by Buan, Marsh and Vatne.
