Related papers: On maximal green sequence for quivers arising from…

Let $B(\Lambda)$ be a level $\ell$ highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\gB(\Lambda)$ of the crystal $B(\Lambda)$ using quiver…

The Green-Griffiths-Lang conjecture says that for every complex projective algebraic variety $X$ of general type there exists a proper algebraic subvariety of $X$ containing all nonconstant entire holomorphic curves $f:\mathbb{C} \to X$.…

Let ${\rm SI}_\beta(Q)$ be the semi-invariant ring of $\beta$-dimensional representations of a quiver $Q$. Suppose that $(Q,\beta)$ projects to another quiver with dimension vector $(Q',\beta')$ through an exceptional representation $E$. We…

We study maximum antichains in two posets related to quiver representations. Firstly, we consider the set of isomorphism classes of indecomposable representations ordered by inclusion. For various orientations of the Dynkin diagram of type…

We study the notion of positive and negative complexity of pairs of objects in cluster categories. The first main result shows that the maximal complexity occurring is either one, two or infinite, depending on the representation type of the…

We give an explicit description of the mutation classes of quivers of type \tilde{A}_n. Furthermore, we provide a complete classification of cluster tilted algebras of type \tilde{A}_n up to derived equivalence. We show that the bounded…

For a given complex projective variety, the existence of entire curves is strongly constrained by the positivity properties of the cotangent bundle. The Green-Griffiths-Lang conjecture stipulates that entire curves drawn on a variety of…

For a rooted cluster algebra $\mathcal{A}(Q)$ over a valued quiver $Q$, a \emph{symmetric cluster variable} is any cluster variable belonging to a cluster associated with a quiver $\sigma (Q)$, for some permutation $\sigma$. The subalgebra…

Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\mathbb{Q}^{ab}$ be the maximal abelian extension of $\mathbb{Q}$. In this article we classify the groups that can arise as $E(\mathbb{Q}^{ab})_{\text{tors}}$ up to…

Let k be an algebraically closed field and A a finite-dimensional k-algebra. Given an A-module M, the set G_e(M) of all submodules of M with dimension vector e is called a quiver Grassmannian. If D,Y are A-modules, then we consider Hom(D,Y)…

For every quiver (valued) of finite representation type we define a finitely presented group called a picture group. This group is very closely related to the cluster theory of the quiver. For example, positive expressions for the Coxeter…

Let $\rm{coh}\mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\rm{coh}\mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left…

A quiver mutation loop is a sequence of mutations and vertex relabelings, along which a quiver transforms back to the original form. For a given mutation loop, we introduce a quantity called a partition q-series. The partition q-series are…

The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type $A_n$, by counting the mutation class of any quiver with underlying graph $A_n$. It will also follow that if $T$ and…

The "No Gap Conjecture" of Br\"ustle-Dupont-P\'erotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two…

We provide a geometric-combinatorial model for the category of coherent sheaves on the weighted projective line of type (2,2,n) via a cylindrical surface with n marked points on each of its upper and lower boundaries, equipped with an order…

In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are…

We construct tilting modules over Jacobian algebras arising from knots. To a two-bridge knot $L[a_1,\ldots,a_n]$, we associate a quiver $Q$ with potential and its Jacobian algebra $A$. We construct a family of canonical indecomposable…

Let $B(\Lambda_0)$ be the level 1 highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\mathbb{B}(\Lambda_0)$ of $B(\Lambda_0)$ via quiver…