McKay Quiver Research Articles

Dihedral $G$-Hilb via representations of the McKay quiver

For a given finite small binary dihedral group $G\subset\mathrm{GL}(2,\mathbf{C})$ we provide an explicit description of the minimal resolution $Y$ of the singularity $\mathbf{C}^{2}/G$. The minimal resolution $Y$ is known to be either the moduli space of $G$-clusters $G$-Hilb$(\mathbf{C}^{2})$, or the equivalent $\mathcal{M}_{\theta}(Q,R)$, the moduli space of $\theta$-stable quiver representations of the McKay quiver. We use both moduli approaches to give an explicit open cover of $Y$, by assigning to every distinguished $G$-graph $\Gamma$ an open set $U_{\Gamma}\subset\mathcal{M}_{\theta}(Q,R)$, and calculating the explicit equation of $U_{\Gamma}$ using the McKay quiver with relations $(Q,R)$.

On McKay quivers and covering spaces

In this paper, we study the relationship between the McKay quivers of finite subgroups of special linear groups and of general linear groups, via natural extension and embedding. We show that under certain condition, the McKay quiver of a finite subgroup G of GL( m ;C) is a regular covering of the McKay quiver of its normal subgroup G ∩ SL( m ;C), and when embedding G in a canonical way into SL( m + 1;C), the new McKay quiver is obtained by adding an arrow from the Nakayama translation of i back to i for each vertex i . We also show by examples that certain interesting McKay quivers can be obtained in these ways.

THE SPECIAL McKAY CORRESPONDENCE AS AN EQUIVALENCE OF DERIVED CATEGORIES

We give a new moduli construction of the minimal resolution of the singularity of type 1/r(1,a) by introducing the Special McKay quiver. To demonstrate that our construction trumps that of the G-Hilbert scheme, we show that the induced tautological line bundles freely generate the bounded derived category of coherent sheaves on X by establishing a suitable derived equivalence. This gives a moduli construction of the Special McKay correspondence for abelian subgroups of GL(2).

On the McKay quivers and m-Cartan matrices

In this paper, we introduce the m-Cartan matrix and observe that some properties of the quadratic form associated to the Cartan matrix of an Euclidean diagram can be generalized to the m-Cartan matrix of a McKay quiver. We also describe the McKay quiver for a finite abelian subgroup of a special linear group.

Triangulated categories of Gorenstein cyclic quotient singularities

We prove there is an equivalence of derived categories between Orlov's triangulated category of singularities for a Gorenstein cyclic quotient singularity and the derived category of representations of a quiver with relations, which is obtained from a McKay quiver by removing one vertex and half of the arrows. This result produces examples of distinct quivers with relations which have equivalent derived categories of representations.

Moduli of McKay quiver representations I: the coherent component

For a finite abelian group G ⊂ GL (n, k ), we describe the coherent component Yθ of the moduli space ℳθ of θ-stable McKay quiver representations. This is a not-necessarily-normal toric variety that admits a projective birational morphism Y θ → A k n / G obtained by variation of Geometric Invariant Theory quotient. As a special case, this gives a new construction of Nakamura's G-Hilbert scheme HilbG that avoids the (typically highly singular) Hilbert scheme of |G|-points in A k n . To conclude, we describe the toric fan of Yθ and hence calculate the quiver representation corresponding to any point ofYθ.

Moduli of McKay quiver representations II: Gröbner basis techniques

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 Y θ 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 representations corresponding to the torus orbits of Y θ , and, in the case where Y θ equals Nakamura's G-Hilbert scheme, we present explicit equations for a cover by local coordinate charts. The latter theorem corrects the first result from Nakamura [I. Nakamura, Hilbert schemes of abelian group orbits, J. Algebraic Geom. 10 (4) (2001) 757–779]. The techniques introduced here allow experimentation in this subject and give concrete algorithmic tools to tackle further open questions. To illustrate this point, we present an example of a nonnormal G-Hilbert scheme, thereby answering a question raised by Nakamura.

Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks

Given an algebraically closed field k of characteristic p≥3, we classify the finite algebraic k-groups whose algebras of measures afford a principal block of tame representation type. The structure of such a group $\mathcal{G}$ is largely determined by a linearly reductive subgroup scheme $\hat{\mathcal{G}}$ of SL(2), with the McKay quiver of $\hat{\mathcal{G}}$ relative to its standard module being the Gabriel quiver of the principal block $\mathcal{B}_0(\mathcal{G})$ . The graphs underlying these quivers are extended Dynkin diagrams of type $\tilde{A}, \tilde{D}$ or $\tilde{E}$ , and the tame blocks are Morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type.

A relation between moduli space of D-branes on orbifolds and Ising model

We study D-branes transverse to an abelian orbifold 3/n × n. The moduli space of the gauge theory on the D-branes is analyzed by combinatorial calculation based on toric geometry. It is shown that the calculation is related to a problem to count the number of ground states of an antiferromagnetic Ising model. The lattice on which the Ising model is defined is a triangular one defined on the McKay quiver of the orbifold.

G2quivers

We present, in explicit matrix representation and a modernity befitting the community, the classification of the finite discrete subgroups of G_2 and compute the McKay quivers arising therefrom. Of physical interest are the classes of N=1 gauge theories descending from M-theory and of mathematical interest are possible steps toward a systematic study of crepant resolutions to smooth G_2 manifolds as well as generalised McKay Correspondences. This writing is a companion monograph to hep-th/9811183 and hep-th/9905212, wherein the analogues for Calabi-Yau three- and four-folds were considered.

ALGEBRA PAIRS ASSOCIATED TO McKAY QUIVERS

ABSTRACT It is known that preprojective algebras are related to the McKay quiver of the finite subgroups of . In this note we relate two algebras to the McKay quiver of a finite subgroup of for any . Some properties of the McKay quiver are also studied with these algebras.

Boundary rings and [formula omitted] coset models

We investigate boundary states of N=2 coset models based on Grassmannians Gr( n, n+ k), and find that the underlying intersection geometry is given by the fusion ring of U( n). This is isomorphic to the quantum cohomology ring of Gr( n, n+ k+1), which in turn can be encoded in a “boundary” superpotential whose critical points correspond to the boundary states. In this way the intersection properties can be represented in terms of a soliton graph that forms a generalized, Z n+k+1 symmetric McKay quiver. We investigate the spectrum of bound states and find that the rational boundary CFT produces only a small subset of the possible quiver representations.

D-branes, exceptional sheaves and quivers on Calabi–Yau manifolds: from Mukai to McKay

We present a method based on mutations of helices which leads to the construction (in the large-volume limit) of exceptional coherent sheaves associated with the (∑ a l a =0) orbits in Gepner models. This is explicitly verified for a few examples including some cases where the ambient weighted projective space has singularities not inherited by the Calabi–Yau hypersurface. The method is based on two conjectures which lead to the analog, in the general case, of the Beilinson quiver for P n . We discuss how one recovers the McKay quiver using the gauged linear sigma model (GLSM) near the orbifold or Gepner point in Kähler moduli space.

McKay Quivers and Extended Dynkin Diagrams

Let $k$ be an algebraically closed field and $G$ a finite nontrivial group whose order is not divisible by the characteristic of $k$. Associated with an $m$-dimensional representation of $G$ is the McKay quiver, whose vertices correspond to the irreducible representations of $G$. We show that if $m = 2$, then the underlying graph of the separated McKay quiver is a finite union of extended Dynkin diagrams.

McKay quivers and extended Dynkin diagrams

Let k k be an algebraically closed field and G G a finite nontrivial group whose order is not divisible by the characteristic of k k . Associated with an m m -dimensional representation of G G is the McKay quiver, whose vertices correspond to the irreducible representations of G G . We show that if m = 2 m = 2 , then the underlying graph of the separated McKay quiver is a finite union of extended Dynkin diagrams.

Rational singularities and almost split sequences

The main aim of this paper is to relate almost split sequences to singularity theory by showing that the McKay quiver built from the finite-dimensional representations of a finite subgroup G G of GL ⁡ ( 2 , C ) \operatorname {GL} (2,{\mathbf {C}}) , where C {\mathbf {C}} is the complex numbers, is isomorphic to the A R AR quiver of the reflexive modules of the quotient singularity associated with G G .