Related papers: Cluster algebras are Cox rings

We construct common triangular bases for almost all the known (quantum) cluster algebras from Lie theory. These bases provide analogs of the dual canonical bases, long anticipated in cluster theory. In cases where the generalized Cartan…

Motivated by Conway and Coxeter's combinatorial results concerning frieze patterns, we sketch an introduction to the theory of cluster algebras and cluster categories for acyclic quivers. The goal is to show how these more abstract theories…

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…

Let $G$ be a compact connected Lie group and $K$ a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of $G$ and $K$ is invertible in a given principal ideal domain $k$. It is known that in…

We study skew-symmetrizable cluster algebras $\mathcal{A}$ associated with unpunctured surfaces $\tilde{\mathbf{S}}$ endowed with an orientation-preserving involution $\sigma$. We give a geometric realization of such cluster algebras by…

We show coordinate rings of open Richardson varieties are upper cluster algebras for any symmetrizable Kac--Moody type. We further show the coordinate rings of (generalized) open Richardson varieties on the twisted product of flag varieties…

Motivated by work of Barot, Geiss and Zelevinsky, we study a collection of Z-bases (which we call companion bases) of the integral root lattice of a root system of simply-laced Dynkin type. Each companion basis is associated with the quiver…

We show that C*-algebras generated by irreducible representations of finitely generated nilpotent groups satisfy the universal coefficient theorem of Rosenberg and Schochet. This result combines with previous work to show that these…

We study criteria for a ring - or more generally, for a small category - to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to…

Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\leq 2$. We construct a triangulated category $\Cc_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$.…

In 1998, Goresky, Kottwitz, and MacPherson showed that for certain spaces X equipped with a torus action, the T-equivariant cohomology ring of X can be described by combinatorial data obtained from its orbit decomposition. Thus, their…

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected-Hodge-type; such schemes should include all…

Let $Q$ be a finite acyclic valued quiver. We give the cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$. These formulas can…

We show that the coordinate ring of an open positroid variety coincides with the cluster algebra associated to a Postnikov diagram. This confirms conjectures of Postnikov, Muller--Speyer, and Leclerc, and generalizes results of Scott and…

Motivated by the theory of cluster algebras, F. Chapoton, S. Fomin and A. Zelevinsky associated to each finite type root system a simple convex polytope called \emph{generalized associahedron}. They provided an explicit realization of this…

Let $G$ be a connected reductive algebraic group. Let $\mathcal{E}\rightarrow \mathcal{B}$ be a principal $G\times G$-bundle and $X$ be a regular compactification of $G$. We describe the Grothendieck ring of the associated fibre bundle…

Given a certain triangulation of a punctured surface with boundary, we construct a new triangulated surface without punctures which covers it. This new surface is naturally equipped with an action of a group of order two, and its quotient…

For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are…

For a ring $R$, we construct a universal $K_R$-torsor $\mathcal{T}_R\to K_{Tate(R)}$ on the $K$-theory space of Tate $R$-modules. This torsor is closely related to canonical central extensions of loop groups. Just like classical loop group…

We introduce Coxeter-sortable elements of a Coxeter group W. For finite W, we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. We characterize Coxeter-sortable elements in…