On complexity of representations of quivers

Injective objects in the category of finitely presented representations of an interval finite quiver

We characterize the indecomposable injective objects in the category of finitely presented representations of an interval finite quiver.

The double of representations of Cohomological Hall Algebra for $A_1$-quiver

We compute two representations of COHA for $A_1$-quiver. The two untwisted representations can be combined into a representation of $D_{n+1}$ Lie algebra. The untwisted increasing representation and the twisted decreasing representation can be combined into a representation of a finite Clifford algebra.

有理箭图表示：驯顺型与野型

Rational quiver representations: tame and wild

该研究针对特征0域上的$K$-有理箭图，利用其Brauer障碍平凡的性质开展分析，并构造出野表示型箭图的实例，实现了给定伽罗瓦扩张的全部Brauer障碍，补充了有理箭图表示理论的相关内容。

Exceptional representations of quivers

Let Q be a connected directed quiver with n vertices. We show that Q is representation-infinite if and only if there do exist n isomorphism classes of exceptional modules of some fixed length at least 2.

Principal Components along Quiver Representations

Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections, and are eigenvectors of an associated matrix pencil.

Orthogonal and symplectic bundles on curves and quiver representations

We show how quiver representations and their invariant theory natu- rally arise in the study of some moduli spaces parametrizing bundles dened on an algebraic curve, and how they lead to ne results regarding the geometry of these spaces.

Representation type of finite quiver Hecke algebras of type $C^{(1)}_{\ell}$

We give a graded dimension formula described in terms of combinatorics of Young diagrams and a simple criterion to determine the representation type for the finite quiver Hecke algebras of type $C_{\ell}^{(1)}$.

Generic representations of wild quivers

We show that any generic representation of a wild connected directed quiver is the union of its subrepresentations of finite length which are regular. As a consequence, we see that the direct limit closure of the preprojective component does not contain any generic module.

A class of perverse sheaves on framed representation varieties of the Jordan quiver

A class of perverse sheaves on framed representation varieties of the Jordan quiver is defined. Its relationship with product of symmetric groups, tensor product of Schur algebras, and tensor product of Fock spaces are addressed.

Moduli of representations of quivers

An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is motivated, and the construction of moduli spaces is reviewed. Topological, arithmetic and algebraic methods for the study of moduli spaces are discussed.

Indecomposable representations for extended Dynkin quivers

We describe a method for an explicit determination of indecomposable preprojective and preinjective representations for extended Dynkin quivers by vector spaces and matrices. This method uses tilting theory and the explicit knowledge of indecomposable modules over the corresponding canonical algebra of domestic type.

广义线程箭图的范畴

Categories of generalized thread quivers

本研究聚焦线程箭图及其商的表示范畴，证明其逐点有限维表示的不可分解性诱导关系，分类内射投射不可分解表示，刻画相关范畴表示性质并构造新的遗传阿贝尔范畴。

Semi-invariants of filtered quiver representations with at most two pathways

A pathway from one vertex of a quiver to another is a reduced path. We modify the classical definition of quiver representations and we prove that semi-invariant polynomials for filtered quiver representations come from diagonal entries if and only if the quiver has at most two pathways between any two vertices. Such class of quivers includes finite $ADE$-Dynkin quivers, affine $\widetilde{A}\widetilde{D}\widetilde{E}$-Dynkin quivers, star-shaped and comet-shaped quivers. Next, we explicitly write all semi-invariant generators for filtered quiver representations for framed quivers with at most…

Representation type of finite quiver Hecke algebras of type $D^{(2)}_{\ell+1}$

We give Erdmann-Nakano type theorem for the finite quiver Hecke algebras $R^{Λ_0}(β)$ of affine type $D^{(2)}_{\ell+1}$, which tells their representation type. If $R^{Λ_0}(β)$ is not of wild representation type, we may compute its stable Auslander-Reiten quiver.

On the recursive construction of indecomposable quiver representations

For a fixed root of a quiver, it is a very hard problem to construct all or even only one indecomposable representation with this root as dimension vector. We investigate two methods which can be used for this purpose. In both cases we get an embedding of the category of representations of a new quiver into the category of representations of the original one which increases dimension vectors. Thus it can be used to construct indecomposable representations of the original quiver recursively. Actually, it turns out that there is a huge class of representations which can be constructed using thes…

Universal Deformation Rings and Quivers of Finite Representation Type

Let k be a field of arbitrary characteristic and let Q be a quiver of finite representation type. In this paper we prove that if M is an indecomposable kQ-module then the universal deformation ring of M over kQ is isomorphic to k.

Etingof conjecture for quantized quiver varieties II: affine quivers

We study the representation theory of quantizations of Gieseker moduli spaces. Namely, we prove the localization theorems for these algebras, describe their finite dimensional representations and two-sided ideals as well as their categories O in some special cases. We apply this to prove our conjecture with Bezrukavnikov on the number of finite dimensional irreducible representations of quantized quiver varieties for quivers of affine type.

Representations of Thread Quivers

We introduce thread quivers as an (infinite) generalization of quivers, and show that every k-linear (k algebraically closed) hereditary category with Serre duality and enough projectives is equivalent to the category of finitely presented representations of a thread quiver. In this way, we obtain an explicit construction of a new class of hereditary categories with Serre duality.

箭图表示的完全可分解子范畴

Completely-decomposable subcategories of quiver representations

针对多参数持续模中驯顺型或野生型箭图表示难以分解的问题，本文提出可计算区分对象的完全可分解子范畴框架。

On the Auslander-Reiten quiver of the representations of an infinite quiver

Let Q be a strongly locally finite quiver and denote by rep(Q) the category of locally finite dimensional representations of Q over some fixed field k. The main purpose of this paper is to get a better understanding of rep(Q) by means of its Auslander-Reiten quiver. To achieve this goal, we define a category C(Q) which is a full, abelian and Hom-finite subcategory of rep(Q) containing all the almost split sequences of rep(Q). We give a complete description of the Auslander-Reiten quiver of C(Q) by describing its connected components. Finally, we prove that these connected components are also c…

Non-commutative Rank and Semi-stability of Quiver Representations

Fortin and Reutenauer defined the non-commutative rank for a matrix with entries that are linear functions. The non-commutative rank is related to stability in invariant theory, non-commutative arithmetic circuits, and Edmonds' problem. We will generalize the non-commutative rank to the representation theory of quivers and define non-commutative Hom and Ext spaces. We will relate these new notions to King's criterion for $σ$-stability of quiver representations, and the general Hom and Ext spaces studied by Schofield. We discuss polynomial time algorithms that compute the non-commutative Homs a…

Unitary and Euclidean representations of a quiver

A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations in canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish t…

有限域上箭图的半稳定表示计数

Counting Semistable Representations of Quivers over Finite Fields

本文推导了有限域上任意箭图具有固定维数向量的绝对不可分解半稳定表示的同构类数的闭式公式，推广了Hua的Kac多项式公式。

Orthoscalar representations of quivers on the category of Hilberts spaces. II

As it is known, finitely presented quivers correspond to Dynkin graphs (Gabriel, 1972) and tame quivers -- to extended Dynkin graphs (Donovan and Freislich, Nazarova, 1973). In the article "Locally scalar reresentations of graphs in the category of Hilbers spaces" (Func. Anal. and Appl., 2005) authors showed the way to tranfer these results to Hilbert spaces, constructed Coxeter functors and proved an analogue of Gabriel theorem fol locally scalar (orthoscalar in the sequel) representations (up to the unitary equivalence). The category of orthoscalar representations of a quiver can be consider…

Projective objects in the category of pointwise finite dimensional representations of an interval finite quiver

For an interval finite quiver $Q$, we introduce a class of flat representations. We classify the indecomposable projective objects in the category $\mathrm{rep}(Q)$ of pointwise finite dimensional representations. We show that an object in $\mathrm{rep}(Q)$ is projective if and only if it is a direct sum of countably generated flat representations.

Quivers of type A, flag varieties and representation theory

A short informal survey on the topics listed in the title. For the proceedings of the International Conference on Rings and Algebras X.

Stability and Indecomposability of the Representations of Quivers of $A_n$-type

In his paper \cite{MR1}, Markus Reineke proposed a conjecture that there exists a stable weight system $Θ$ for every indecomposable representation of Dynkin type quiver. In this paper, we showed this conjecture is true for quivers of $A_n$-type by combinatorial construction of a special weight system. We also reinterpret this weight system in terms of semi-invariant theory.

Enhanced Nilpotent Representations of a Cyclic Quiver

We define a set of "enhanced" nilpotent quiver representations that generalizes the enhanced nilpotent cone. This set admits an action by an associated algebraic group $K$ with finitely many orbits. We define a combinatorial set that parametrizes the set of orbits under this action and we derive a purely combinatorial formula for the dimension of an orbit.

Representation type of finite quiver Hecke algebras of type $A^{(1)}_{\ell}$ for arbitrary parameters

We give Erdmann-Nakano type theorem for the finite quiver Hecke algebras $R^{Λ_0}(β)$ of affine type $A^{(1)}_{\ell}$. Note that each finite quiver Hecke algebra lies in one parameter family, and the original Erdmann-Nakano theorem studied the finite quiver Hecke algebra at a special parameter value. We study the general case in our paper. Our result shows in particular that their representation type does not depend on the parameter. Moreover, when the parameter value is nonzero, we show that finite quiver Hecke algebras of tame representation type are biserial algebras.

类型 $A$ 连续箭图的最大刚性表示

Maximal rigid representations of continuous quivers of type $A$

本文研究类型 $A$ 连续箭图的最大刚性表示，针对 Igusa、Rock 和 Todorov 引入的对象，给出了其同构类数量的计算公式。

The Representation Theory of Dynkin Quivers. Three Contributions

These notes provide three contributions to the (well-established) representation theory of Dynkin and Euclidean quivers. They should be helpful as part of a direct approach to study representations of quivers, and they may shed some new light on properties of Dynkin and Euclidean quivers. Part 1 deals with the case A (see arXiv:1304.5720). Part 2 concerns the case D. We show that the category of representation of D_n contains a full subcategory which is equivalent to the category of representations of a quiver of type D_{n-1} such that the remaining indecomposable representations are thin. Thi…

A characterization of representation infinite quiver settings

We characterize pairs (Q,d) consisting of a quiver Q and a dimension vector d, such that over a given algebraically closed field k there are infinitely many representations of Q of dimension vector d. We also present an application of this result to the study of algebras with finitely many orbits with respect to the action of (the double product) of the group of units.

Representations of quivers and mixed graphs

This is a survey article for "Handbook of Linear Algebra", 2nd ed., Chapman & Hall/CRC, 2014. An informal introduction to representations of quivers and finite dimensional algebras from a linear algebraist's point of view is given. The notion of quiver representations is extended to representations of mixed graphs, which permits one to study systems of linear mappings and bilinear or sesquilinear forms. The problem of classifying such systems is reduced to the problem of classifying systems of linear mappings.

等维箭图表示及其 $U$-不变量

Equidimensional quiver representations and their $U$-invariants

本文研究等维箭图表示的 $U$-不变量，提出通过为每个顶点选取关联箭的映射来构造不变量域的自由生成元系统及相应截面。

Generalized quiver Hecke algebras

We generalize the geometric construction of quiver Hecke algebras from Varagnolo and Vasserot to a setup with arbitrary connected reductive groups. This corresponds to replacing quiver representations by generalized quiver representations introduced by Derksen and Weyman. The class of algebras which we construct contains (affine) nil Hecke algebras, skew group rings of Weyl groups with polynomial rings and quiver Hecke algebras. We describe an explicit faithful representation in a polynomial ring and we calculate the generators and relations for these algebras.

GIT-Equivalence and Semi-Stable Subcategories of Quiver Representations

In this paper, we answer the question of when the subcategory of semi-stable representations is the same for two rational vectors for an acyclic quiver. This question has been previously answered by Ingalls, Paquette, and Thomas in the tame case in [10]. Here we take a more invariant theoretic approach, to answer this question in general. We recover the known result in the tame case.

Geometric approach to Hall algebra of representations of Quivers over local ring

By using perverse sheaves on representation spaces of quivers over $k[t]/(t^n)$ and jet schemes over flag varieties, we construct a geometric composition algebra $\mathbf K$ under Lusztig's framework on geometric realizations of the negative part of quantum algebras. Simple perverse sheaves in $\mathbf K$ form the canonical basis of $\mathbf K$. The relationships among the algebra $\mathbf K$, the composition algebra of locally projective representations of quivers over $k[t]/(t^n)$ and quantum generalized Kac-Moody algebra are provided.

Counting Representations of Quivers Respecting Nilpotent Relations over Finite Fields

This paper presents analogous results of Hua [7][8] on numbers of representations of quivers over finite fields which respect nilpotent relations under certain assumptions. A closed formula which counts isomorphism classes of absolutely indecomposable representations with given dimension vectors is given and a $q$-deformation of Weyl-Kac denominator identity is established. In principle, if the numbers of representations are known, then the numbers of isomorphism classes of absolutely indecomposable representations are known.

Connectivity Properties of McKay Quivers

We present several results regarding the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay quivers, as well as quivers that cannot arise as McKay quivers, and discuss a necessary and sufficient condition for two finite groups to share a connected McKay quiver.

On quiver Grassmannians and orbit closures for representation-finite algebras

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.

Representations of Quiver Hecke Algebras via Lyndon Bases

A new class of algebras have been introduced by Khovanov and Lauda and independently by Rouquier. These algebras categorify one-half of the Quantum group associated to arbitrary Cartan data. In this paper, we use the combinatorics of Lyndon words to construct the irreducible representations of those algebras associated to Cartan data of finite type. This completes the classification of simple modules for the quiver Hecke algebra initiated by Kleshchev and Ram.

Double Flag Varieties and Representations of Quivers

We gave a classification of P and Q with a finite number of K-orbits of a double flag variety G/P*K/Q for a symmetric pair (G, K) when G=GL_{m+n} and K=GL_{m}*GL_{n}, and a description of K-orbits when the number of K-orbits of G/P*K/Q is finite. We solved the problem by providing a correspondence between the K-orbits and the quiver representations.

无限箭图的局部有限维表示的Gabriel定理

Gabriel's Theorem for Locally Finite-Dimensional Representations of Infinite Quivers

本文证明了无限箭图的局部有限维表示的Gabriel定理，指出连通箭图的局部有限维表示范畴具有唯一表示型当且仅当其基础图为广义ADE Dynkin图。

Quiver representations of constant Jordan type and vector bundles

Inspired by the work of Benson, Carlson, Friedlander, Pevtsova, and Suslin on modules of constant Jordan type for finite group schemes, we introduce in this paper the class of representations of constant Jordan type for an acyclic quiver $Q$. We do this by first assigning to an arbitrary finite-dimensional representation of $Q$ a sequence of coherent sheaves on moduli spaces of thin representations. Next, we show that our quiver representations of constant Jordan type are precisely those representations for which the corresponding sheaves are locally free. We also construct representations of …

On the Auslander-Reiten quiver for the category of representations of partially ordered sets with an involution

In this article we describe the Auslander-Reiten quiver for some posets with an involution, that we call types $\mathfrak{U}_n$ and $\mathfrak{U}_\infty$. These posets appear in the differentiation III of Zavadskij [12]. We follow the approach to classical Auslander-Reiten theory due to Auslander, Reiten and Smalø[1]. For this purpose, we give a natural exact structure for the category of representations of a partially ordered set with an involution. We describe the projective and injective representations

Quiver representations of maximal rank type and an application to representations of a quiver with three vertices

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 with three vertices. From this construction it follows that real root representations of this quiver are tree modules. Moreover, formulae given by Ringel can be applied to compute the dimension of the e…

Representation type via Euler characteristics and singularities of quiver Grassmannians

In this text, we characterize the representation type of an acyclic quiver by the properties of its associated quiver Grassmannians. This characterization utilizes and extends known results about singular quiver Grassmannians and cell decompositions into affine spaces. While all quiver Grassmannians for indecomposable representations of quivers of finite representation types $A$ and $D$ are smooth and admit cell decompositions, it turns out that all quiver Grassmannians for indecomposable representations of quivers of tame types $A$ and $D$ admit cell decompositions, but some of these quiver G…

Quiver representations respecting a quiver automorphism: a generalisation of a theorem of Kac

We consider quiver representations respecting a quiver automorphism and show that the dimension vectors of the indecomposables are precisely the positive roots of an associated symmetrisable Kac-Moody Lie algebra. Moreover, every such Lie algebra arises in this way. We also discuss the relationship with species of valued quivers over finite fields.

Invariants of mixed representations of quivers I

We introduce a new concept of mixed representations of quivers that is a generalization of ordinary representations of quivers and orthogonal (symplectic) representations of symmetric quivers introduced recently by Derksen and Weyman. We describe the generating invariants of mixed representations of quivers (First Fundamental Theorem) and prove additional results that allow us to describe the defining relations between them in the second article.

Identities for mixed representations of quiver are finitely based

The classical construction of representations of quivers enables us to consider linear maps between several vector spaces. The mixed representations of quivers helps us to work with linear maps as well as bilinear forms on several vector spaces. We construct a finite generating set for the T-ideal of identities of the polynomial invariants of mixed representations of a quiver.

Quiver representations in abelian categories

We introduce the notion of (twisted) quiver representations in abelian categories and study the category of such representations. We construct standard resolutions and coresolutions of quiver representations and study basic homological properties of the category of representations. These results are applied in the case of parabolic vector bundles and framed coherent sheaves.

Dimension expanders via quiver representations

We relate the notion of dimension expanders to quiver representations and their general subrepresentations, and use this relation to establish sharp existence results.

Orbit semigroups and the representation type of quivers

We show that a finite, connected quiver Q without oriented cycles is a Dynkin or Euclidean quiver if and only if all orbit semigroups of representations of Q are saturated.

Decompositions of Infinite-Dimensional $A_{ \infty,\infty}$ Quiver Representations

Using linear algebraic methods we show that every (possibly infinite-dimensional) representation of a quiver with underlying graph $A_{\infty, \infty}$ is Krull-Schmidt, as long as the arrows in the quiver eventually point outward.