Related papers: Compatibility degree of cluster complexes
Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional…
We study $f$-vectors, which are the maximal degree vectors of $F$-polynomials in cluster algebra theory. For a cluster algebra is of finite type, we find that positive $f$-vectors correspond with $d$-vectors, which are exponent vectors of…
Identifying clusters of vertices in graphs continues to be an important problem, and modularity continues to be used as a tool for solving the problem. Modularity, which measures the quality of a division of the vertices into clusters,…
We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category $\mathcal{C}$ and cluster-tilting subcategory $\mathcal{T}$ satisfying only mild finiteness conditions. We show that the structure theory…
The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of…
We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation…
We construct geometric realization for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on…
We present a generator of random networks where both the degree-dependent clustering coefficient and the degree distribution are tunable. Following the same philosophy as in the configuration model, the degree distribution and the…
For most networks, the connection between two nodes is the result of their mutual affinity and attachment. In this paper, we propose a mutual selection model to characterize the weighted networks. By introducing a general mechanism of…
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…
We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different…
Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose…
A general notion of a quasi-finite algebra is introduced as an algebra graded by the set of all integers equipped with topologies on the homogeneous subspaces satisfying certain properties. An analogue of the regular bimodule is introduced…
The denominator conjecture, proposed by Fomin and Zelevinsky, says that for a cluster algebra, the cluster monomials are uniquely determined by their denominator vectors with respect to an initial cluster. In this paper, for a cluster…
A compatible associative algebra is a vector space endowed with two associative multiplication operations that satisfy a natural compatibility condition. In this paper, we investigate and classify compatible pairs of associative algebras of…
Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G. In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B. We also prove that every…
Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type A cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial…
Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer…
We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent…
We introduce admissible group actions on cluster algebras, cluster categories and quivers with potential and study the resulting orbit spaces. The orbit space of the cluster algebra has the structure of a generalized cluster algebra. This…