Related papers: Polyhedral models for generalized associahedra via…
M. Carr and S. Devadoss introduced in [7] the notion of tubing on a finite simple graph $\Gamma$, in the context of configuration spaces on the Hilbert plane. To any finite simple graph $\Gamma$ they associated a finite partially ordered…
We introduce a rich family of generalizations of the pentagram map sharing the property that each generates an infinite configuration of points and lines with four points on each line. These systems all have a description as $Y$-mutations…
We initiate a systematic study of cohomology theories for partial groups, algebraic structures introduced by Chermak that generalize groups by allowing only partially defined products. Inspired by classical group cohomology, we develop two…
We give an elementary construction of polyhedra whose links are connected bipartite graphs, which are not necessarily isomorphic pairwise. We show, that the fundamental groups of some of our polyhedra contain surface groups. In particular,…
Based on the construction of polytope functions and several results about them in [LP], we take a deep look on their mutation behaviors to find a link between a face of a polytope and a sub-cluster algebra of the corresponding cluster…
We extend a factorization theorem by Gwo\'zdziewicz and Hejmej from the ring of formal power series to any complete regular local ring $ R $. More precisely, let $ f \in R $ and assume that its Newton polyhedron has a loose edge such that…
We continue our investigation on denominator conjecture of Fomin and Zelevinsky for cluster algebras via geometric models initialed in \cite{FG22}. In this paper, we confirm the denominator conjecture for cluster algebras of finite type.…
We realize a family of generalized cluster algebras as Caldero-Chapoton algebras of quivers with relations. Each member of this family arises from an unpunctured polygon with one orbifold point of order 3, and is realized as a…
In this paper, we will answer one of the questions proposed by Bodur, Del~Pia, Dey, Molinaro and Pokutta in 2017. Specifically, we show that the k-aggregation closure of a covering set is a polyhedron. The proof technique is based on an…
It has been established in recent years how to approach acyclic cluster algebras of finite type using subword complexes. In this paper, we continue this study by describing the c- and g-vectors, and by providing a conjectured description of…
We consider the Ptolemy cluster algebras, which are cluster algebras of finite type $A$ (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed $\zS$ in a…
This work presents the tessellations and polytopes from the perspective of both n-dimensional geometry and abstract symmetry groups. It starts with a brief introduction to the terminology and a short motivation. In the first part, it…
There exist natural generalizations of the real moduli space of Riemann spheres based on manipulations of Coxeter complexes. These novel spaces inherit a tiling by the graph-associahedra convex polytopes. We obtain explicit configuration…
We explain how the approach of Andre and Quillen to defining cohomology and homology as suitable derived functors extends to generalized (co)homology theories, and how this identification may be used to study the relationship between them.…
This is a concise introduction to Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition of cluster algebras (geometric, without coefficients), construct…
We establish certain fundamental properties of $f$-vectors and $F$-matrices for generalized cluster algebras, including the initial and final seed mutation formulas, the compatibility property and the symmetry property. Along the way, we…
We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was previously announced in arXiv:1201.0031. Mongardi showed that the subgroup…
Broline, Crowe and Isaacs have computed the determinant of a matrix associated to a Conway-Coxeter frieze pattern. We generalise their result to the corresponding frieze pattern of cluster variables arising from the Fomin-Zelevinsky cluster…
We show that for cluster algebras associated with finite quivers without oriented cycles (with no coefficients), a seed is determined by its cluster, as conjectured by Fomin and Zelevinsky.We also obtain an interpretation of the monomial in…
The homogeneous coordinate ring $\mathbb{C}[\operatorname{Gr}(k,n)]$ of the Grassmannian is a cluster algebra, with an additive categorification $\operatorname{CM}C$. Thus every $M\in\operatorname{CM}C$ has a cluster character…