Quiver Grassmannians Research Articles

On Cluster Variables of Rank Two Acyclic Cluster Algebras

In this note, we find an explicit formula for the Laurent expression of cluster variables of coefficient-free rank two cluster algebras associated with the matrix \({{\left(\begin{array}{ll} 0 &c \\ -c &0 \end{array}\right)}}\), which manifestly shows that a large number of coefficients are non-negative. As a corollary, we obtain an explicit expression for the Euler-Poincaré characteristics of the corresponding quiver Grassmannians.

Quantum cluster variables via Serre polynomials

For skew-symmetric acyclic quantum cluster algebras, we express the quantum F-polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, we obtain the existence of counting polynomials for these varieties and the positivity conjecture with respect to acyclic seeds. These results complete previous work by Caldero and Reineke and confirm a recent conjecture by Rupel.

Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras

We study quiver Grassmannians associated with indecomposable representations of the Kronecker quiver. We find a cellular decomposition of them and we compute their Betti numbers. As an application, we give a geometric realization of the "canonical basis" of cluster algebras of Kronecker type (found by Sherman and Zelevinsky) and of type $A_2^{(1)}$.

A Homological Interpretation of the Transverse Quiver Grassmannians

In recent articles, the investigation of atomic bases in cluster algebras associated to affine quivers led the second-named author to introduce a variety called transverse quiver Grassmannian and the first-named and third-named authors to consider the smooth loci of quiver Grassmannians. In this paper, we prove that, for any affine quiver Q, the transverse quiver Grassmannian of an indecomposable representation M is the set of points N in the quiver Grassmannian of M such that Ext^1(N,M/N)=0. As a corollary we prove that the transverse quiver Grassmannian coincides with the smooth locus of the irreducible components of minimal dimension in the quiver Grassmannian.

Generic bases for cluster algebras and the Chamber Ansatz

Let Q Q be a finite quiver without oriented cycles, and let Λ \Lambda be the corresponding preprojective algebra. Let g \mathfrak {g} be the Kac-Moody Lie algebra with Cartan datum given by Q Q , and let W W be its Weyl group. With w ∈ W w \in W , there is associated a unipotent cell N w N^w of the Kac-Moody group with Lie algebra g \mathfrak {g} . In previous work we proved that the coordinate ring C [ N w ] \mathbb {C}[N^w] of N w N^w is a cluster algebra in a natural way. A central role is played by generating functions φ X \varphi _X of Euler characteristics of certain varieties of partial composition series of X X , where X X runs through all modules in a Frobenius subcategory C w \mathcal {C}_w of the category of nilpotent Λ \Lambda -modules. The first aim of this article is to compare the function φ X \varphi _X with the so-called cluster character of X X , which is defined in terms of the Euler characteristics of quiver Grassmannians. We show that for every X X in C w \mathcal {C}_w , φ X \varphi _X coincides, after an appropriate change of variables, with the cluster character of Fu and Keller associated with X X using any cluster-tilting object T T of C w \mathcal {C}_w . A crucial ingredient of the proof is the construction of an isomorphism between varieties of partial composition series of X X and certain quiver Grassmannians. This isomorphism is obtained in a very general setup and should be of interest in itself. Another important tool of the proof is a representation-theoretic version of the Chamber Ansatz of Berenstein, Fomin and Zelevinsky, adapted to Kac-Moody groups. As an application, we get a new description of a generic basis of the cluster algebra A ( Γ _ T ) \mathcal {A}(\underline {\Gamma }_T) obtained from C [ N w ] \mathcal {C}[N^w] via specialization of coefficients to 1. Here generic refers to the representation varieties of a quiver potential arising from the cluster-tilting module T T . For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont.

Quiver grassmannians, quiver varieties and the preprojective algebra

Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are homeomorphic to the lagrangian quiver varieties of Nakajima which have been well studied in the context of geometric representation theory. We then refine this result by finding quiver grassmannians which are homeomorphic to the Demazure quiver varieties introduced by the first author, and others which are homeomorphic to the graded/cyclic quiver varieties defined by Nakajima. The Demazure quiver grassmannians allow us to describe injective objects in the category of locally nilpotent modules of the preprojective algebra. We conclude by relating our construction to a similar one of Lusztig using projectives in place of injectives.

Friezes and a construction of the Euclidean cluster variables

Let Q be a Euclidean quiver. Using friezes in the sense of Assem–Reutenauer–Smith, we provide an algorithm for computing the (canonical) cluster character associated with any object in the cluster category of Q . In particular, this algorithm allows us to compute all the cluster variables in the cluster algebra associated with Q . It also allows us to compute the sum of the Euler characteristics of the quiver Grassmannians of any module M over the path algebra of Q .

Euler Characteristics of Quiver Grassmannians and Ringel-Hall Algebras of String Algebras

We compute the Euler characteristics of quiver Grassmannians and quiver flag varieties of tree and band modules and prove their positivity. This generalizes some results by G. Cerulli Irelli (2010). As an application we consider the Ringel-Hall algebra ${\mathcal C}(A)$ of some string algebras A and compute in combinatorial terms the products of arbitrary functions in ${\mathcal C}(A)$ . These results are transferred to covering theory.

On the cardinalities of Kronecker quiver Grassmannians

We deduce using the Ringel–Hall algebra approach explicit formulas for the cardinalities of some Grassmannians over a finite field associated to the Kronecker quiver. We realize in this way a quantification of the formulas obtained by Caldero and Zelevinsky for the Euler characteristics of these Grassmannians. Finally we present a recursive algorithm for computing the cardinality of every Kronecker quiver Grassmannian over a finite field.

Quiver Grassmannians associated with string modules

We provide a technique to compute the Euler characteristic of a class of projective varieties called quiver Grassmannians. This technique applies to quiver Grassmannians associated with "orientable string modules". As an application we explicitly compute the Euler characteristic of quiver Grassmannians associated with indecomposable preprojective, preinjective and regular homogeneous representations of an affine quiver of type $\tilde{A}_{p,1}$. For $p=1$, this approach provides another proof of a result due to P. Caldero and A. Zelevinsky in \cite{CZ}.

Transverse quiver Grassmannians and bases in affine cluster algebras

Sherman-Zelevinsky and Cerulli constructed canonically positive bases in cluster algebras associated to affine quivers having at most three vertices. Both constructions involve cluster monomials and normalized Chebyshev polynomials of the first kind evaluated at a certain imaginary element in the cluster algebra. Using this combinatorial description, it is possible to define for any affine quiver $Q$ a set $\mathcal B(Q)$ which is conjectured to be the canonically positive basis of the acyclic cluster algebra $\mathcal A(Q)$. In this article, we provide a geometric realization of the elements in $\mathcal B(Q)$ in terms of the representation theory of $Q$. This is done by introducing an analogue of the Caldero-Chapoton cluster character where the usual quiver Grassmannian is replaced by a constructible subset called transverse quiver Grassmannian.

On cluster algebras arising from unpunctured surfaces II

We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler–Poincaré characteristic of quiver Grassmannians in Dynkin type A and affine Dynkin type A ˜ .