This paper classifies a class of holomorphic D-branes, closely related to framed torsion-free sheaves, on threefolds fibered in resolved ADE surfaces over a general curve C, in terms of representations with relations of a twisted Kronheimer– Nakajima-type quiver in the category Coh(C) of coherent sheaves on C. For the local Calabi–Yau case C ∼= A and special choice of framing, one recovers the ...

The notion of ∗-representations of quivers can be obtained from ordinary quiver representations by considering coaction of groups on “vertex” vector spaces together with action. A generating system for the algebra of semiinvariants of ∗-representations of a quiver is determined. This was done by reducing the problem to the case of bipartite quivers of the special form and by introducing a funct...

For physicists: We show that the quiver gauge theory derived from a Calabi-Yau cone via an exceptional collection of line bundles on the base has the original cone as a component of its classical moduli space. For mathematicians: We use data from the derived category of sheaves on a Fano surface to construct a quiver, and show that its moduli space of representations has a component which is is...

In [3] and [13], the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the properties of 2–Calabi–Yau (Auslander–Reiten formula) and high order associativity. Introduction Cluster algebras were introduced by Fomin and Zelevinsky in [9]. By definition, the cluste...

We show that the dimension of the mesh algebra of a finite Auslander-Reiten quiver over a field is a purely combinatorial invariant that does not depend on the ground field. Moreover, a combinatorial algorithm for computing this dimension is given along our proof of this result. Translation quivers appear naturally in the representation theory of finite dimensional algebras; see, for example, [...

The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type An, by counting the mutation class of any quiver with underlying graph An. It will also follow that if T and T ′ are cluster-tilting objects in a cluster category C, then EndC(T ) is isomorphic to EndC(T ) if and only if T = τ T . 1. Cluster-tilted algebras The cluster categ...

In this paper we introduce several computational techniques for the study of moduli spaces of McKay quiver representations, making use of Gröbner bases and toric geometry. For a finite abelian group G ⊂ GL(n, k), let Yθ be the coherent component of the moduli space of θ-stable representations of the McKay quiver. Our two main results are as follows: we provide a simple description of the quiver...

The descent algebra Σ(W) is a subalgebra of the group algebra QW of a finite Coxeter group W, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of W. Thus Σ(W) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct Σ(W) as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean ...

We analyse the classical moduli spaces of supersymmetric vacua of 3d N = 2 Chern-Simons quiver gauge theories. We show quite generally that the moduli space of the 3d theory always contains a baryonic branch of a parent 4d N = 1 quiver gauge theory, where the 4d baryonic branch is determined by the vector of 3d Chern-Simons levels. In particular, starting with a 4d quiver theory dual to a 3-fol...