Generalizing Schubert cells in type A and a cell decomposition if Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of an oriented cycle admit an affine cell decomposition parametrized by multi-tableaux. We show that they carry a torus operation and describe the T -equivariant cohomology using GoreskyKottwitz-MacPherson-theory. A...

The Auslander-Reiten quiver of a finite-dimensional associative algebra A encodes information about the indecomposable finite-dimensional representations of A and their homomorphisms. A component of the AuslanderReiten quiver is called preprojective if it does not admit oriented cycles and each of its modules can be shifted into a projective module using the AuslanderReiten translation. Preproj...

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 Yθ → Ak /G obtained by variation of GIT quotient. As a special case, this gives a new construction of Nakamura’s G-Hilbert scheme Hilb that avoids the (typically...

We introduce the notion of “maximal rank type” for representations of quivers, which requires certain collections of maps involved in the representation to be of maximal rank. We show that real root representations of quivers are of maximal rank type. By using the maximal rank type property and universal extension functors we construct all real root representations of a particular wild quiver w...

In this paper we present simplifying techniques which allow one to compute the quiver diagrams for various D-branes at (non-Abelian) orbifold singularities with and without discrete torsion. The main idea behind the construction is to take the orbifold of an orbifold. Many interesting discrete groups fit into an exact sequence N → G → G/N . As such, the orbifold M/G is easier to compute as (M/N...

We construct geometric realizations of the r-colored bosonic and fermionic Fock space on the equivariant cohomology of the moduli space of framed rank r torsion-free sheaves on CP 2. Using these constructions, we realize geometrically all level one irreducible representations of the affine Lie algebra gl(r). The cyclic group Z k acts naturally on the moduli space of sheaves, and the fixed-point...

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 la = 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 CalabiYau hypersurface. The method is based o...

Toric Duality arises as an ambiguity in computing the quiver gauge theory living on a D3-brane which probes a toric singularity. It is reviewed how, in simple cases Toric Duality is Seiberg Duality. The set of all Seiberg Dualities on a single node in the quiver forms a group which is contained in a larger group given by a set of Picard-Lefschetz transformations. This leads to elements in the g...

Given a maximal rigid object T of the cluster tube, we determine the objects finitely presented by T . We then use the method of Keller and Reiten to show that the endomorphism algebra of T is Gorenstein and of finite representation type, as first shown by Vatne. This algebra turns out to be the Jacobian algebra of a certain quiver with potential, when the characteristic of the base field is no...

Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig’s quiver varieties. This gives a geometric realization of the crystal graph of the lower half of the quantum group associated to a simply-laced Kac-Moody algebra. Using an enumeration of the irreducible components of Lusztig’s quiver varieties in finite and affine type A by combinatorial data, w...