Related papers: Generalized cluster complexes via quiver represent…

We give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of perfect matchings of angles. Our formula simplifies the cluster expansion formula given by…

We study finite quasi-quantum groups in their quiver setting developed recently by the first author in arXiv:0902.1620 and arXiv:0903.1472. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation…

Generalized Cluster Algebras (GCA) are generalizations of Cluster Algebras (CA) with higher-order exchange relations. Previously, Chekhov-Shapiro conjectured that every GCA can be embedded into a CA. In this paper, we prove a modified…

Cluster categories were introduced in 2006 by Buan-Marsh-Reineke-Reiten-Todorov in order to categorify acyclic cluster algebras without coefficients. Their construction was generalized by Amiot (2009) and Plamondon (2011) to arbitrary…

This is a survey article for "Handbook of Linear Algebra", 2nd ed., Chapman & Hall/CRC, 2014. An informal introduction to representations of quivers and finite dimensional algebras from a linear algebraist's point of view is given. The…

Using a representation theoretic parameterization for the orbits in the enhanced cyclic nilpotent cone, derived by the authors in a previous article, we compute the fundamental group of these orbits. This computation has several…

The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field FF_q and any sequence ii of simple objects in C the element X_{V,ii}…

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees. We show that the number of isomorphism classes of such trees is given by the Catalan number Cn where n is…

We introduce thread quivers as an (infinite) generalization of quivers, and show that every k-linear (k algebraically closed) hereditary category with Serre duality and enough projectives is equivalent to the category of finitely presented…

In this paper, we show that every regular singular $\mathscr{D}$-module in $\mathbb{C}^n$ whose singular locus is a normal crossing is isomorphic to a quiver $\mathscr{D}$-module - a $\mathscr{D}$-module whose definition is based on certain…

We study generalized cluster algebras introduced by Chekhov and Shapiro. When the coefficients satisfy the normalization and quasi-reciprocity conditions, one can naturally extend the structure theory of seeds in the ordinary cluster…

Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such…

The $F$-triangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the $F$-triangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the…

We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of `extended quivers' which are oriented hypergraphs. We describe mutations of such…

The irreducible modules over quiver Hecke superalgebras $R_\theta$ can be classified in terms of cuspidal modules. To an indivisible positive root $\alpha$ and a non-negative integer $d$, one associates a quotient $\bar R_{d\alpha}$ of…

We first provide an explicit combinatorial description of the Auslander-Reiten quiver $\Gamma^Q$ of finite type $D$. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra…

Let $\Phi$ be a finite root system of rank $n$ and let $m$ be a nonnegative integer. The generalized cluster complex $\Delta^m (\Phi)$ was introduced by S. Fomin and N. Reading. It was conjectured by these authors that $\Delta^m (\Phi)$ is…

We construct the moduli space of finite dimensional representations of generalized quivers for arbitrary connected complex reductive groups using Geometric Invariant Theory as well as Symplectic reduction methods. We explicit characterize…

We employ projective Fra\"iss\'e theory to define the "generic combinatorial $n$-simplex" as the pro-finite, simplicial complex that is canonically associated with a family of simply defined selection maps between finite triangulations of…

We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the…