Quiver Representations Research Articles

Desingularization of quiver Grassmannians for Dynkin quivers

A desingularization of arbitrary quiver Grassmannians for representations of Dynkin quivers is constructed in terms of quiver Grassmannians for an algebra derived equivalent to the Auslander algebra of the quiver.

Localization in quiver moduli spaces

Torus fixed points of quiver moduli spaces are given by stable representations of the universal (abelian) covering quiver. As far as the Kronecker quiver is concerned they can be described by stable representations of certain bipartite quivers coming along with a stable colouring. By use of the glueing method it is possible to construct a huge class of such quivers implying a lower bound for the Euler characteristic. For certain roots it is even possible to construct all torus fixed points.

On the Harder-Narasimhan filtration for finite dimensional representations of quivers

We prove that the Harder-Narasimhan filtration for an unstable finite dimensional representation of a finite quiver coincides with the filtration associated to the 1-parameter subgroup of Kempf, which gives maximal unstability in the sense of Geometric Invariant Theory for the corresponding point in the parameter space where these objects are parametrized in the construction of the moduli space.

COTORSION THEORY IN THE CATEGORY OF QUIVER REPRESENTATIONS

This note is devoted to the study of cotorsion theory in the category of quiver representations. Let [Formula: see text] be a quiver, ξ be a class of representations, and let Vξ denote the class of all modules each of which appears in a vertex of an element of ξ. If the pair (⊥Vξ, Vξ) (respectively, (Vξ, Vξ⊥)) is a cotorsion theory, we investigate conditions under which the pair (⊥ξ, ξ) (respectively, (ξ, ξ⊥)) is a cotorsion theory and vice versa. We show that in certain cases, completeness can be transferred between these cotorsion theories.

On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants

Abstract In previous work we have shown that the equivariant index of multi-centered $ \mathcal{N}=2 $ black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription — the Coulomb branch formula — for computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke’s result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincaré polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package “CoulombHiggs.m” implementing the Coulomb and Higgs branch formulae.

Kac's Theorem for equipped graphs and for maximal rank representations

AbstractWe give two generalizations of Kac's Theorem on representations of quivers. One is to representations of equipped graphs by relations, in the sense of Gelfand and Ponomarev. The other is to representations of quivers in which certain of the linear maps are required to have maximal rank.

PURE INJECTIVE REPRESENTATIONS OF QUIVERS

Let R be a ring and Q be a quiver. In this paper we give an- other definition of purity in the category of quiver representations. Under such definition we prove that the class of all pure injective representations of Q by R-modules is preenveloping. In case Q is a left rooted semi-co- barren quiver and R is left Noetherian, we show that every cotorsion flat representation of Q is pure injective. If, furthermore, R is n-perfect and F is a flat representation Q, then the pure injective dimension of F is at most n.

Filtering free resolutions

AbstractA recent result of Eisenbud–Schreyer and Boij–Söderberg proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. When does this numerical decomposition correspond to an actual filtration of the minimal free resolution? Our main result gives a sufficient condition for this to happen. We apply it to show the non-existence of free resolutions with some plausible-looking Betti diagrams and to study the semigroup of quiver representations of the simplest ‘wild’ quiver.

Stacks and Torsors over Quivers

We study natural Grothendieck topologies on categories of quivers without and with relations, prove descent theorems for quiver representations, and introduce the notion of torsors over quivers.

Transfinite Tree Quivers and their Representations

The idea of "vertex at the infinity" naturally appears when studying indecomposable injective representations of tree quivers. In this paper we formalize this behavior and find the structure of all the indecomposable injective representations of a tree quiver of size an arbitrary cardinal $\kappa$. As a consequence the structure of injective representations of noetherian $\kappa$-trees is completely determined. In the second part we will consider the problem whether arbitrary trees are source injective representation quivers or not.

Quantum groups via Hall algebras of complexes

We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a nite eld Hall algebra construction involving Z2-graded complexes of quiver representations.

Framed moduli and Grassmannians of submodules

In this work we study a realization of moduli spaces of framed quiver representations as Grassmannians of submodules devised by Marcus Reineke. Obtained is a generalization of this construction for finite dimensional associative algebras and for quivers with oriented cycles. As an application we get an explicit realization of fibers for the moduli space bundle over the categorical quotient for the quiver $A_{n-1}^{(1)}$.

Quiver polynomials in iterated residue form

Degeneracy loci polynomials for quiver representations generalize several important polynomials in algebraic combinatorics. In this paper we give a nonconventional generating sequence description of these polynomials, when the quiver is of Dynkin type.

The minimal model program for the Hilbert scheme of points on [formula omitted] and Bridgeland stability

In this paper, we study the birational geometry of the Hilbert scheme P2[n] of n-points on P2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n≤9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions.

Orbit closures and rank schemes

Let A be a finitely generated associative algebra over an algebraically closed field k , and consider the variety \mathrm{mod}_A^d(k) of A -module structures on k^d . In case A is of finite representation type, equations defining the closure \bar{\mathcal O}_M are known for M \in \mathrm{mod}_A^d(k) ; they are given by rank conditions on suitable matrices associated with M . We study the schemes \mathcal{C}_M defined by such rank conditions for modules over arbitrary A , comparing them with similar schemes defined for representations of quivers and obtaining results on singularities. One of our main theorems is a description of the ideal of \bar{\mathcal O}_M for a representation M of a quiver of type \mathbb{A}_n , a result Lakshmibai and Magyar established for the equioriented quiver of type \mathbb{A}_n in [12].

Finitistic Dimension Conjectures for representations of quivers

Let R be a ring and Q be a quiver. We prove the first Finitistic Dimension Conjecture to be true for RQ, the path ring of Q over R, provided that R satisfies the conjecture. In fact, we prove that if the little and the big finitistic dimensions of R coincide and equal n

Compactifications of spaces of Landau–Ginzburg models

This paper reviews the results and techniques from the authors' work on Landau–Ginzburg degenerations and applies them in basic examples. The main example is the -category, where we observe a relationship to stability conditions and directed quiver representations. We conclude with a brief survey of applications to the birational geometry of del Pezzo surfaces.

Infinitely many $ \mathcal{N} $ = 2 SCFT with ADE flavor symmetry

We present evidence that for each ADE Lie group G there is an infinite tower of 4D $ \mathcal{N} $ = 2 SCFTs, which we label as D(G, s) with s ∈ $ \mathbb{N} $ , having (at least) flavor symmetry G. For G = SU(2), D(SU(2), s) coincides with the Argyres-Douglas model of type D s+1, while for larger flavor groups the models are new (but for a few previously known examples). When its flavor symmetry G is gauged, D(G, s) contributes to the Yang-Mills beta-function as $ \frac{s}{{2\left( {s+1} \right)}} $ adjoint hypermultiplets. The argument is based on a combination of Type IIB geometric engineering and the categorical deconstruction of arXiv:1203.6743 . One first engineers a class of $ \mathcal{N} $ = 2 models which, trough the analysis of their category of quiver representations, are identified as asymptotically-free gauge theories with gauge group G coupled to some conformal matter system. Taking the limit g YM → 0 one isolates the matter SCFT which is our D(G, s).

Quiver bialgebras and monoidal categories

We study the bialgebra structures on quiver coalgebras and the monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed monoidal categories.

On the Euler characteristic of Kronecker moduli spaces

Combining the MPS degeneration formula for the Poincare polynomial of moduli spaces of stable quiver representations and localization theory, it turns that the determination of the Euler characteristic of these moduli spaces reduces to a combinatorial problem of counting certain trees. We use this fact in order to obtain an upper bound for the Euler characteristic in the case of the Kronecker quiver. We also derive a formula for the Euler characteristic of some of the moduli spaces appearing in the MPS degeneration formula.