Quiver Representations Research Articles

Generalized quivers, orthogonal and symplectic representations, and Hitchin–Kobayashi correspondences

We review the theory of quiver bundles over a Kähler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group [Formula: see text]. We first study the case when [Formula: see text] or [Formula: see text], interpreting them as orthogonal (respectively symplectic) bundle representations of the symmetric quivers introduced by Derksen–Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. In particular, we completely characterize the polystable forms of such representations. Finally, we discuss Hitchin–Kobayashi correspondences for these objects.

On Generic Semi-simple Decomposition of Dimension Vector for an Arbitrary Quiver

Generic (canonical) decomposition of dimension vector for a quiver was introduced by Victor Kac as characterizing the generic module indecomposable summands dimensions, hence, the generic orbit. Derksen and Weyman proposed an elegant algorithm to compute that decomposition, extensively using Schofield’s results. We consider generic semi-simple decomposition, which corresponds to generic closed orbit and provide a simple and fast algorithm to compute this decomposition. Generic semi-simple decomposition has two useful application. First, it reduces the computation of generic decomposition to the case of quiver without oriented cycles in a geometric way. Second, it provides a nice novel presentation of the algebra of invariants of quiver representations as a tensor product of similar algebras for the summands.

Completeness of the induced cotorsion pairs in categories of quiver representations

Given a complete hereditary cotorsion pair (A,B) in an abelian category C satisfying certain conditions, we study the completeness of the induced cotorsion pairs (Φ(A),Φ(A)⊥) and (Ψ⊥(B),Ψ(B)) in the category Rep(Q,C) of C-valued representations of a given quiver Q. We show that if Q is left rooted, then the cotorsion pair (Φ(A),Φ(A)⊥) is complete, and if Q is right rooted, then the cotorsion pair (Ψ⊥(B),Ψ(B)) is complete. Besides, we work on the infinite line quiver A∞∞, which is neither left rooted nor right rooted. We prove that these cotorsion pairs in Rep(A∞∞,R) are complete, as well.

Combinatorics of Exceptional Sequences in Type A

Exceptional sequences are certain sequences of quiver representations. We introduce a class of objects called strand diagrams and use these to classify exceptional sequences of representations of a quiver whose underlying graph is a type $\mathbb{A}_n$ Dynkin diagram. We also use variations of these objects to classify $c$-matrices of such quivers, to interpret exceptional sequences as linear extensions of explicitly constructed posets, and to give a simple bijection between exceptional sequences and certain saturated chains in the lattice of noncrossing partitions.

Group actions on quiver varieties and applications

We study algebraic actions of finite groups of quiver automorphisms on moduli spaces of quiver representations. We decompose the fixed loci using group cohomology and we give a modular interpretation of each component. As an application, we construct branes in hyperkähler quiver varieties, as fixed loci of such actions.

Tree Normal Forms for Quiver Representations

We explore methods for constructing normal forms of indecomposable quiver representations. The first part of the paper develops homological tools for recursively constructing families of indecomposable representations from indecomposables of smaller dimension vector. This is then specialized to the situation of tree modules, where the existence of a special basis simplifies computations and gives nicer normal forms. Motivated by a conjecture of Kac, we use this to construct cells of indecomposable representations as deformations of tree modules. The second part of the paper develops geometric tools for constructing cells of indecomposable representations from torus actions on moduli spaces of representations. As an application, we combine these methods to construct families of indecomposables grouped into cells. These actually give a normal form for all indecomposables of certain roots.

Topological strings, strips and quivers

We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, Gromov-Witten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver. This has various deep consequences; in particular, expressing open BPS invariants in terms of motivic Donaldson-Thomas invariants, immediately proves integrality of the former ones. Taking advantage of the relation to quivers we also derive explicit expressions for classical open BPS invariants for an arbitrary strip geometry, which lead to a large set of number theoretic integrality statements. Furthermore, for a specific framing, open topological string partition functions for strip geometries take form of generalized q-hypergeometric functions, which leads to a novel representation of these functions in terms of quantum dilogarithms and integral invariants. We also study quantum curves and A-polynomials associated to quivers, various limits thereof, and their specializations relevant for strip geometries. The relation between toric manifolds and quivers can be regarded as a generalization of the knots-quivers correspondence to more general Calabi-Yau geometries.

Tiered trees, weights, and q-Eulerian numbers

Maxmin trees are labeled trees with the property that each vertex is either a local maximum or a local minimum. Such trees were originally introduced by Postnikov [12], who gave a formula to count them and different combinatorial interpretations for their number. In this paper we generalize this construction and define tiered trees by allowing more than two classes of vertices. Tiered trees arise naturally when counting the absolutely indecomposable representations of certain quivers, and also when one enumerates torus orbits on certain homogeneous varieties. We define a notion of weight for tiered trees and prove bijections between various weight 0 tiered trees and other combinatorial objects; in particular order n weight 0 maxmin trees are naturally in bijection with permutations on n−1 letters. We conclude by using our weight function to define a new q-analogue of the Eulerian numbers.

The (cyclic) enhanced nilpotent cone via quiver representations

The {{,mathrm{GL},}}(V)-orbits in the enhanced nilpotent cone Vtimes mathcal {N} are (essentially) in bijection with the orbits of a certain parabolic Psubseteq {{,mathrm{GL},}}(V) (the mirabolic subgroup) in the nilpotent cone mathcal {N}. We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameterization generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different from the previous ones that have appeared in the literature. Explicit translations between the different parametrizations are given.

Parallels between Moduli of Quiver Representations and Vector Bundles over Curves

This is a review article exploring similarities between moduli of quiver representations and moduli of vector bundles over a smooth projective curve. After describing the basic properties of these moduli problems and constructions of their moduli spaces via geometric invariant theory and symplectic reduction, we introduce their hyperk\"ahler analogues: moduli spaces of representations of a doubled quiver satisfying certain relations imposed by a moment map and moduli spaces of Higgs bundles. Finally, we survey a surprising link between the counts of absolutely indecomposable objects over finite fields and the Betti cohomology of these (complex) hyperk\"ahler moduli spaces due to work of Crawley-Boevey and Van den Bergh and Hausel, Letellier and Rodriguez-Villegas in the quiver setting, and work of Schiffmann in the bundle setting.

Quiver representations and their applications

In this article, we survey some results on geometric methods to study quiver representations, and applications of these results to sheaves, equivariant sheaves and parabolic bundles.

On the Structural Theorem of Persistent Homology

We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the Matrix Structural Theorem. Any of the various algorithms for computing persistent homology constitutes a constructive proof of the Matrix Structural Theorem. We show that the Matrix Structural Theorem is equivalent to the Krull–Schmidt property of the category of filtered chain complexes. We separately establish the Krull–Schmidt property by abstract categorical methods, yielding a novel nonconstructive proof of the Matrix Structural Theorem. These results provide the foundation for an alternate categorical framework for decomposition in persistent homology, bypassing the usual persistence vector spaces and quiver representations.

Geometry of moduli space of sheaves on del Pezzo surfaces via Maruyama transform

In this paper, we prove that the moduli space of stable pairs (or sheaves) on del Pezzo surface S is isomorphic to the Maruyama type transformation of a projective bundle of a quiver representation space.

Stratifications for moduli of sheaves and moduli of quiver representations

We study the relationship between two stratifications on parameter spaces for coherent sheaves and for quiver representations: a stratification by Harder–Narasimhan types and a stratification arising from the geometric invariant theory construction of the associated moduli spaces of semistable objects. For quiver representations, both stratifications coincide, but this is not quite true for sheaves. We explain why the stratifications on various Quot schemes do not coincide and that the correct parameter space to compare such stratifications is the stack of coherent sheaves, where we construct an asymptotic GIT stratification and prove that this coincides with the Harder–Narasimhan stratification. Then we relate these stratifications for sheaves and quiver representations using a generalisation of a construction of Alvarez-Consul and King.

Book Review: Elementary applied topology; Persistence theory: From quiver representations to data analysis

Book Review: Elementary applied topology; Persistence theory: From quiver representations to data analysis

Multiplications and eigenvalues of tensors via linear maps

ABSTRACTIn this paper, we study multiplications and eigenvalues of tensors by using linear maps. By the composition of linear maps, we naturally obtain two multiplications of tensors, which generalize Shao's product, Einstein product and n-mode product etc. Some properties of product of tensors are obtained directly, which don't need proofs, for example the associative law of Shao's product etc. Moreover, we extend eigenvalues of tensors to tensor representations of quivers. By using tensor representations of quivers, eigenvalues of tensors and properties are given.

Mixed cobinary trees

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees (MCTs). We show that the number of isomorphism classes of such trees is given by the Catalan number [Formula: see text] where [Formula: see text] is the number of internal nodes. We also consider the corresponding quiver [Formula: see text] of type [Formula: see text]. As a special case of more general known results about the relation between [Formula: see text]-vectors, representations of quivers and their semi-invariants, we explain the bijection between MCTs and the vertices of the generalized associahedron corresponding to the quiver [Formula: see text]. These results are extended to [Formula: see text]-clusters in the next paper. We give one application: a new short proof of a conjecture of Reineke using MCTs.

Nori Motives of Curves With Modulus and Laumon 1-motives

AbstractLet k be a number field. We describe the category of Laumon 1-isomotives over k as the universal category in the sense of M. Nori associated with a quiver representation built out of smooth proper k-curves with two disjoint effective divisors and a notion of for such “curves with modulus”. This result extends and relies on a theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's Abelian category of motives.

A Universal Investigation of $n$-representations of n-quivers

noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $mathit{Q}_1$,$mathit{Q}_2$,..., $mathit{Q}_n$, we construct a new quiver $mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}}$, called an $n$-quiver, and identify each category $Rep_k(mathit{Q}_j)$ of representations of a quiver $mathit{Q}_j$ as a full subcategory of the category $Rep_k(mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}})$ of representations of $mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}}$ for every $j in {1,2,ldots , n}$.

Cohomological orientifold Donaldson–Thomas invariants as Chow groups

We establish a geometric interpretation of orientifold Donaldson–Thomas invariants of $$\sigma $$ -symmetric quivers with involution. More precisely, we prove that the cohomological orientifold Donaldson–Thomas invariant is isomorphic to the rational Chow group of the moduli space of $$\sigma $$ -stable self-dual quiver representations. As an application we prove that the Chow Betti numbers of moduli spaces of stable m-tuples in classical Lie algebras can be computed numerically. We also prove a cohomological wall-crossing formula relating semistable Hall modules for different stabilities.