Related papers: A refined multiplication formula for cluster chara…
This paper presents a unified matrix factorization framework for classical and robust clustering. We begin by revisiting the well-known equivalence between crisp k-means clustering and matrix factorization, following and rigorously…
Markov numbers, i.e. positive integers appearing in solutions to $x^2 + y^2 + z^2 = 3xyz$, can be viewed as specializations of cluster variables. The second author and Matsushita gave a generalization of the Markov equation, $x^2 + y^2 +…
We describe a reduction technique for stably 2-Calabi--Yau Frobenius extriangulated categories $\mathcal{F}$ with respect to a functorially finite rigid subcategory $\mathcal{X}$. The reduction of such a category is another category…
We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields…
Motivated by cluster ensembles, we introduce a new variant of frieze patterns associated to acyclic cluster algebras, which we call ${\bf Y}\textit{-frieze patterns}$. Using the mutation rules for ${\bf Y}$-variables, we define a large…
We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay…
We study abelian localizations of triangulated categories induced by rigid contravariantly finite subcategories, and also triangulated structures on subfactor categories of triangulated categories. In this context we generalize recent…
We present a combinatorial model for cluster algebras of type $D_n$ in terms of centrally symmetric pseudotriangulations of a regular $2n$-gon with a small disk in the centre. This model provides convenient and uniform interpretations for…
The preprints arXiv:math/0610728 and arXiv:math/0612451 are withdrawn due to a problem with Theorem 2.2 in arXiv:math/0610728. The theorem claims that for certain triangulated categories with finitely many indecomposable objects, the…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…
The article gives a ring theoretic perspective on cluster algebras. Gei{\ss}-Leclerc-Schr\"oer prove that all cluster variables in a cluster algebra are irreducible elements. Furthermore, they provide two necessary conditions for a cluster…
We show the polynomial property of $F$-polynomials for generalized quantum cluster algebras and obtain the associated separation formulas under a mild condition. Along the way, we obtain Gupta's formulas of $F$-polynomials for generalized…
A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but subclasses have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties.…
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…
We prove a formula for double Schubert and Grothendieck polynomials specialized to two rearrangements of the same set of variables. Our formula generalizes the usual formulas for Schubert and Grothendieck polynomials in terms of RC-graphs,…
We build foundations of an approach to study canonical forms of $2$-Calabi--Yau triangulated categories with cluster-tilting objects, using dg algebras and relative singularity categories. This is motivated by cluster theory, singularity…
We put cluster tilting in ageneral framework by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal one-orthogonal subcategory) carries an abelian structure. These abelian quotients turn out…
Classification of cluster variables in cluster algebras (in particular, Grassmannian cluster algebras) is an important problem, which has direct application to computations of scattering amplitudes in physics. In this paper, we apply the…
We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a…
A Calabi-Yau pair of index one and complexity zero is toric. Furthermore, a Calabi-Yau pair of index one and complexity one is of cluster type. In this article, we study Calabi-Yau pairs of index one and complexity two. We develop machinery…