Related papers: A cluster structure on the coordinate ring of part…

We introduce a framework for $\mathbb{Z}$-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a…

We present here two detailed examples of additive categorifications of the cluster algebra structure of a coordinate ring of a maximal unipotent subgroup of a simple Lie group. The first one is of simply-laced type ($A_3$) and relies on an…

By work of a number of authors, beginning with Scott and culminating with Galashin and Lam, the coordinate rings of positroid varieties in the Grassmannian carry cluster algebra structures. In fact, they typically carry many such…

Let $D$ be the Auslander algebra of $\mathbb{C}[t]/(t^n)$, which is quasi-hereditary, and $\mathcal{F}_\Delta$ the subcategory of good $D$-modules. For any $\mathsf{J}\subseteq[1, n-1]$, we construct a subcategory…

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ be its two opposite Borel subgroups. For two elements $u$, $v$ of the Weyl group $W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the…

For a finite-dimensional simple Lie algebra $\mathfrak{g}$ admitting a non-trivial minuscule representation and a connected marked surface $\Sigma$ with at least two marked points and no punctures, we prove that the cluster algebra…

Let $S$ be an upper cluster algebra, which is a subalgebra of $R$. Suppose that there is some cluster variable $x_e$ such that ${R}_{{x}_e} = S[{x}_e^{\pm 1}]$. We try to understand under which conditions ${R}$ is an upper cluster algebra,…

We introduce new objects, called $(G,c)$-bands, associated with a simple simply-connected algebraic group $G$, and a Coxeter element $c$ in its Weyl group. We show that bands of a given type are the $K$-points of an infinite dimensional…

For a classical group $G$ and a Coxeter element $c$ of the Weyl group, it is known that the coordinate ring $\mathbb{C}[G^{e,c^2}]$ of the double Bruhat cell $G^{e,c^2}:=B\cap B_-c^2B_-$ has a structure of cluster algebra of finite type,…

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $B_r$, $B$ and $B_-$ be its two opposite Borel subgroups, and $W$ be the associated Weyl group. For $u$, $v\in W$, it is known that the coordinate ring ${\mathbb…

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always…

We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings.…

We describe a ring whose category of Cohen-Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a…

Let $\widehat G \subseteq G$ be complex reductive algebraic groups. The branching problem that aims to study $G$-modules as $\widehat G$-modules is encoded by a collection of branching multiplicities parameterised by pairs of dominant…

We prove the Berenstein-Zelevinsky conjecture that the quantized coordinate rings of the double Bruhat cells of all finite dimensional simple algebraic groups admit quantum cluster algebra structures with initial seeds as specified by [4].…

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 prototypical examples of a cluster algebra is the coordinate ring of a finite Grassmannian: using the Pl\"ucker embedding the cluster algebra structure allows one to move between `maximal sets' of algebraically independent Pl\"ucker…

We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig's coordinates. Several…

Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point…

In this paper, we prove one case of the conjecture given by Hernandez and Leclerc\cite{HL0}. Specifically, we give a cluster algebra structure on the Grothendieck ring of a full subcategory of the finite dimensional representations of a…