Dirk Kussin, Hagen Meltzer
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.
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We show that for a Dynkin quiver $Q$ of type $E_7$ with a specific orientation, the path algebra $KQ$ has no slope function of the form $μ=\fracθ{\dim}$ that defines a total stability condition. This gives a counterexample to a conjecture of Reineke.
Let $Q$ be a connected quiver with no oriented cycles, $k$ the field of complex numbers and $P$ a projective representation of $Q$. We study the adjoint action of the automorphism group $\Aut_{kQ} P$ on the space of radical endomorphisms $\radE_{kQ}P$. Using generic equivalence, we show that the quiver $Q$ has the property that there exists a dense open $\Aut_{kQ} P$-orbit in $\radE_{kQ} P$, for all projective representations $P$, if and only if $Q$ is a Dynkin quiver. This gives a new characterisation of Dynkin quivers.
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 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.
We classify the singularities in the unframed Nakajima quiver varieties associated with extended Dynkin quivers and the corresponding minimal imaginary root with a small restriction on the parameter and use this to construct a number of hyper-Kähler cobordisms between binary polyhedral spaces.
We show that the mutation class of a finite quiver without oriented cycles is finite if and only is the quiver is either Dynkin, extended Dynkin or has at most two vertices.
This paper studies silted algebras, namely, endomorphism algebras of 2-term silting complexes, over path algebras of Dynkin quivers. We will describe an algorithm to produce all basic 2-term silting complexes over the path algebra of a Dynkin quiver, and use this algorithm to compute some examples.
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…
Total stability and Auslander-Reiten theory for Dynkin quivers
本文研究Dynkin箭图的全稳定性问题，提出仅需验证Auslander-Reiten箭图边界处的几乎可裂序列满足斜率不等式即可判定全稳定，简化了全稳定稳定性函数的判定条件。
Let $Q$ be the 3-Kronecker quiver, i.e., $Q$ has two vertices, labeled by 1 and 2, and three arrows from 2 to 1. Fix an algebraically closed field $k$. Let $\mathcal{C}$ be a regular component of the Auslander-Reiten quiver containing an indecomposable module $X$ with dimension $(1,1)$ or $(2,1)$. Using the properties of the Fibonacci numbers, we show that the Gabriel-Roiter measures of the indecomposable modules in $\mathcal{C}$ are uniquely determined by the dimension vectors. In other words, two indecomposable modules in $\mathcal{C}$ are not isomorphic if and only if their Gabriel-Roiter m…
A piecewise-linear formula for rigid representations of Dynkin quivers
该研究推导了给定维数向量的Dynkin箭图刚性表示的分段线性公式，并通过多个示例对公式进行了说明，为Dynkin箭图刚性表示的相关研究提供了新的表达工具。
The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. Given this conjecture, I give a proof of the r…
In this paper, we compute the product structure of the Hochschild cohomology of preprojective algebras of quivers of type D and E over a field of characteristic zero.
For a stability condition $σ$ on a triangulated category, Dimitrov-Katzarkov introduced the notion of a $σ$-exceptional collection. In this paper, we study full $σ$-exceptional collections in the derived category of an acyclic quiver. In particular, we prove that any stability condition $σ$ on the derived category of a Dynkin quiver admits a full $σ$-exceptional collection.
Let k be a field and A the n-Kronecker algebra, this is the path algebra of the quiver with 2 vertices, a source and a sink, and n arrows from the source to the sink. It is well-known that the dimension vectors of the indecomposable A-modules are the positive roots of the corresponding Kac-Moody algebra. Thorsten Weist has shown that for every positive root there are even tree modules with this dimension vector and that for every positive imaginary root there are at least n tree modules. Here, we present a short proof of this result. The considerations used also provide a calculation-free proo…
We use the theory of differential tensor algebras and their modules to produce explicit representations of extended Dynkin quivers.
Degenerations in graded quiver varieties and in derived categories of Dynkin quivers
本文研究了迪金克准旗导出范畴中退化现象，揭示了格雷德准旗变分与导出范畴退化之间的对应关系。
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。
The Hochschild homology/cohomology an associative algebra, together with the Connes differential, the contraction map and the Lie derivative, forms the structure of calculus. In this paper we compute explicitely the calculus structure of preprojective algebras of Dynkin quivers over a field of characteristic zero. This also completes the work in math.AG/0502301, where the Batalin-Vilkovisky structure of the Hochschild cohomology of preprojective algebras of non-Dynkin quivers are computed and the calculus can be easily computed from that.
Let M and N be two representations of an extended Dynkin quiver such that the orbit O_N of N is contained in the orbit closure \bar{O_M} and has codimension two. We show that the pointed variety $(\bar{O_M},N)$ is smoothly equivalent to a simple surface singularity of type A_n, or to the cone over a rational normal curve.
For coprime dimension vectors certain torus fixed points of the Kronecker moduli space are indecomposable tree modules. They are indecomposable representations of the regular m-tree and can be glued in order to get stable torus fixed point for every coprime dimension vector. Using their stability and the reflection functor we show that for arbitrary roots there exist indecomposable tree modules of the Kronecker quiver as factor modules of these torus fixed points.
For Dynkin quivers, we find the Laurent polynomials $\widetilde{X}_{a, c}^{b}(v)$ and use $\widetilde{X}_{a, c}^{b}(v)$ to construct the Hall algebra $\hc_v(\cc(\cp))$ over $\mz[v, v^{-1}]$, where $\widetilde{X}_{a, c}^{b}(|\mf_q|)$'s are structure constants used by Bridgeland. The Laurent polynomials $\widetilde{X}_{a, c}^{b}(v)$ are explicitly given in $A_1$ case. As an application, we obtain the full quantum groups $U_t(\sg)$ associated to the Dynkin quivers for arbitrary $t\not=0,\pm1$.
In this paper we compute b-functions (or Bernstein-Sato polynomials) of various semi-invariants of quivers. The main tool is an explicit relation for the b-functions between semi-invariants that correspond to each other under reflection functors (or castling transforms). This enables us to compute recursively the b-functions for all Dynkin quivers, and extended Dynkin quivers with prehomogeneous dimension vectors.
Extending the main result of Part 1, in the first part of this paper we show that every quiver Grassmannian of a representation of a quiver of extended Dynkin type $D$ has a decomposition into affine spaces. In the case of real root representations of small defect, the non-empty cells are in one-to-one correspondence to certain, so called non-contradictory, subsets of the vertex set of a fixed tree-shaped coefficient quiver. In the second part, we use this characterization to determine the generating functions of the Euler characteristics of the quiver Grassmannians (resp. $F$-polynomials). Al…
Series of Fano quiver moduli from extended Dynkin quivers
该研究针对小型扩展 Dynkin 箭图，构造了皮卡秩≤5的任意偶维 Fano 簇系列，给出其到射影空间的平坦态射及贝蒂数闭式公式，还阐释了上同调霍尔代数对其的作用。
We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type $A$, we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type $A$ quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation.
Flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology
本文证明了Dynkin箭图（A、D、E型）的箭图Grassmannian旗版本无奇上同调，A和D型具仿射铺砌，并推广了刚性表示的对角分解结果。
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…
Classification silted algebras for a quiver of Dynkin type $\mathbb{A}_{n}$ via geometric models
本文利用gentle代数的几何模型完整分类了Dynkin型$\mathbb{A}_n$线性定向箭图的泥沙代数，证明其恰为$A_n$型倾斜代数或$A_m\times A_{n-m}$型倾斜代数，并据此给出计数公式。
We use the geometric technique, developed by Weyman, to calculate the resolution of orbit closures of representations of Dynkin quivers with every vertex being source or sink. We use this resolution to derive the normality of such orbit closures. As a consequence we obtain the normality of certain orbit closures of type E.
Quivers variety for affine orthogonal quiver
本文研究正交箭图的箭图簇，利用预投影代数中的图形演算，确定了由仿射Dynkin图产生的Kleinian曲面奇点类型。
Classification of silted algebras for two quivers of Dynkin type $\mathbb{A}_{n}$
本文完整分类了两类Dynkin型$\mathbb{A}_{n}$箭图的淤泥代数，并计算了其数量。
We study the Grothendieck classes of quiver cycles, i.e. invariant closed subvarieties of the representation space of a quiver. For quivers without oriented loops we show that the class of a quiver cycle is determined by quiver coefficients, which generalize the earlier studied quiver coefficients for equioriented quivers of type A. We conjecture that quiver coefficients satisfy positivity and finiteness properties. Our main result is a formula for the quiver coefficients for orbit closures of Dynkin type with rational singularities, which confirms the finiteness conjecture. This formula is ba…
For a Dynkin quiver $Q$ of type ADE and a sum $β$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of the corresponding type via equivariant K-theory, imitating Ginzburg-Reshetikhin-Vasserot's geometric realization of the quantum affine Schur-Weyl duality. Our construction is based on Hernandez-Leclerc's isomorphism between a certain graded quiver variety and the space of representations of the quiver $Q$ of dimension vector $β$. We identify the functor induced from our bimodule with Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl d…
Let $\mathcal{K}_n$ be the so-called wild Kronecker quiver, i.e., a quiver with one source and one sink and $n\geq 3$ arrows from the source to the sink. The following problems will be studied for an arbitrary regular component $\mathcal{C}$ of the Auslander-Reiten quiver: (1) What is the relationship between dimension vectors and quasi-lengths of the indecomposable regular representations in $\mathcal{C}$? (2) For a given natural number $d$, is there an upper bound of the number of indecomposable representations in $\mathcal{C}$ with the same length $d$? (3) When do the sets of the dimension …
With the help of Lusztig's canonical basis, we study local intersection cohomology of the Zariski closures of orbits of representations of a quiver of type A, D or E. In particular, we characterize the rationally smooth orbits and prove that orbit closures are smooth if and only if they are rationally smooth. This provides an analogue of theorems of V. Deodhar, and J. Carrell and D. Peterson on Schubert varieties.
As a continuation of the previous paper, we find a combinatorial interpretation of Dorey's rule for type $C_n$ via twisted Auslander-Reiten quivers (AR-quivers) of type $D_{n+1}$, which are combinatorial AR-quivers related to certain Dynkin diagram automorphisms. Combinatorial properties of twisted AR-quivers are useful to understand not only Dorey's rule but also other notions in the representation theory of the quantum affine algebra $U_q'(C_n^{(1)})$ such as denominator formulas. In addition, unlike twisted adapted classes of type $A_{2n-1}$ in the previous paper, we show twisted AR-quivers…
We study indecomposable representations of quivers on separable infinite-dimensional Hilbert spaces by bounded operators. We consider a complement of Gabriel's theorem for these representations. Let $Γ$ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams $\tilde{A_n} (n \geq 0)$, $\tilde{D_n} (n \geq 4)$, $\tilde{E_6}$,$\tilde{E_7}$ and $\tilde{E_8}$, then there exists an indecomposable representation of $Γ$ on separable infinite-dimensional Hilbert spaces.
We derive a geometric model for the module category $\operatorname{mod} \mathbf{k} Q$ of a Dynkin quiver $Q$ via the heart of a total stability condition on the bounded derived category of $\operatorname{mod} \mathbf{k} Q$. As an application, we prove Reineke's conjecture that there is a stability function on $\operatorname{mod} \mathbf{k} Q$ making any indecomposables stable.
We introduce and study the twisted adapted $r$-cluster point and its combinatorial Auslander-Reiten quivers, called twisted AR-quivers and folded AR-quivers, of type $A_{2n+1}$ which are closely related to twisted Coxeter elements and the non-trivial Dynkin diagram automorphism. As applications of the study, we prove that folded AR-quivers encode crucial information on the representation theory of quantum affine algebra $U_q'(B^{(1)}_{n+1})$ such as Dorey's rule and denominator formulas.
Inspired by recent work of Cerulli-Feigin-Reineke on desingularizations of quiver Grassmannians of representations of Dynkin quivers, we obtain desingularizations in considerably more general situations and in particular for Grassmannians of modules over iterated tilted algebras of Dynkin type. Our desingularization map is constructed from Nakajima's desingularization map for graded quiver varieties.
Let $\Bbbk$ be an algebraically closed field, $Q$ a finite quiver, and denote by $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$ the affine $\Bbbk$-scheme of representations of $Q$ with a fixed dimension vector ${\mathbf{d}}$. Given a representation $M$ of $Q$ with dimension vector ${\mathbf{d}}$, the set ${\mathcal{O}}_M$ of points in $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}(\Bbbk)$ isomorphic as representations to $M$ is an orbit under an action on $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}(\Bbbk)$ of a product of general linear groups. The orbit ${\mathcal{O}}_M$ and its Zariski closure $\overline{\mathcal{…
Let $Q$ be a quiver of extended Dynkin type $D$. In this first of two papers, we show that the quiver Grassmannian $Gr_e(M)$ has a decomposition into affine spaces for every dimension vector $e$ and every indecomposable representation $M$ of defect $-1$ and defect $0$, with exception of the non-Schurian representations in homogeneous tubes. We characterize the affine spaces in terms of the combinatorics of a fixed coefficient quiver for $M$. The method of proof is to exhibit explicit equations for the Schubert cells of $Gr_e(M)$ and to solve this system of equations successively in linear term…
We study fundamental group of the exchange graphs for the bounded derived category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category D(Γ_N Q) of the Calabi-Yau-N Ginzburg algebra associated to Q. In the case of D(Q), we prove that its space of stability conditions (in the sense of Bridgeland) is simply connected; as applications, we show that its Donanldson-Thomas invariant can be calculated via a quantum dilogarithm function on exchange graphs. In the case of D(Γ_N Q), we show that faithfulness of the Seidel-Thomas braid group action (which is known for Q of type A or N = …
The famous pentagon identity for quantum dilogarithms has a generalization for every Dynkin quiver, due to Reineke. A more advanced generalization is associated with a pair of alternating Dynkin quivers, due to Keller. The description and proof of Keller's identities involves cluster algebras and cluster categories, and the statement of the identity is implicit. In this paper we describe Keller's identities explicitly, and prove them by a dimension counting argument. Namely, we consider quiver representations $\boldsymbol{\mathrm{Rep}}_γ$ together with a superpotential function $W_γ$, and calc…
We use a geometric description of $m$-cluster categories of Dynkin type $A$ to count the the number of coloured quivers in the $m$-mutation class of quivers of Dynkin type $A$. This is related to angulations of polygons and the cell-growth problem.
Classification of silted algebras for two quivers of Dynkin type $\mathbb{D}_{n}$
本文对 Dynkin 型 $\mathbb{D}_{n}$ 的两个箭图（线性定向及其唯一源点反转）完全分类了淤积代数和严格短代数，给出计数公式，证明后者均为串代数，并构造实现函子不延拓为导出等价的例子。
For a Dynkin quiver $Q$ (of type ADE), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc's monoidal category $\mathcal{C}_Q$ of modules over the quantum loop algebra $U_q(L\mathfrak{g})$ via Nakajima's homomorphism. As an application, we show that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives an equivalence between the cate…
Tropical friezes are the tropical analogues of Coxeter-Conway frieze patterns. In this note, we study them using triangulated categories. A tropical frieze on a 2-Calabi-Yau triangulated category $\mathcal{C}$ is a function satisfying a certain addition formula. We show that when $\mathcal{C}$ is the cluster category of a Dynkin quiver, the tropical friezes on ${\mathcal{C}}$ are in bijection with the $n$-tuples in ${\mathbb{Z}}^n$, any tropical frieze $f$ on $\mathcal{C}$ is of a special form, and there exists a cluster-tilting object such that $f$ simultaneously takes non-negative values or …
For any acyclic quiver, we establish a family of structure isomorphisms for its cohomological Hall algebra (CoHA). The family is parameterized by partitions of the quiver into Dynkin subquivers. For each such partition, we write the domain of our isomorphism as a tensor product of subalgebras in two ways. In the first, each tensor factor is isomorphic to the CoHA of the quiver with a single vertex and no arrows. In the second, the tensor factors are each isomorphic to the CoHAs of the corresponding Dynkin subquivers. When the quiver is already an orientation of a simply-laced Dynkin diagram, o…
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.
Any moduli space of representations of a quiver (possibly with oriented cycles) has an embedding as a dense open subvariety into a moduli space of representations of a bipartite quiver having the same type of singularities. A connected quiver is Dynkin or extended Dynkin if and only if all moduli spaces of its representations are smooth.
In this article we study the interplay between algebro-geometric notions related to $π$-points and structural features of the stable Auslander-Reiten quiver of a finite group scheme. We show that $π$-points give rise to a number of new invariants of the AR-quiver on one hand, and exploit combinatorial properties of AR-components to obtain information on $π$-points on the other. Special attention is given to components containing Carlson modules, constantly supported modules, and endo-trivial modules.
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 …
Let $A$ be a finite-dimensional algebra over an algebraically closed field. The problem of constructing indecomposable $A$-modules inductively from simple ones by means of exact sequences - called accessibility - is the starting point of the present diploma-thesis. It has lead us to the consideration of exceptional and indecomposable tree-representations of finite quivers. Following Ringel, we prove his result that exceptional representations are tree-representations. We give a detailed description of the various aspects of the Schofield-Induction which plays an important role in the proof. Mo…
Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay-Cartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKay-Cartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup G. In the special case of the classica…
It is shown that, given a representation of a quiver over a finite field, one can check in polynomial time whether it is absolutely indecomposable.
Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations
本文通过有向图方法构造了A型Dynkin图相关的Yang代数及其矩形表示，并展示了其与Gelfand-Tsetlin基的关联。
Singularities of normalized R-matrices and E-invariants for Dynkin quivers
通过等变反常层丛研究ADE型量子环代数在Hernandez-Leclerc一级子范畴中任意单模间规范化R矩阵的奇异性，证明极点阶数与Dynkin箭图对应装饰表示的E不变量维数相等，建立有限ADE型丛代数加性与幺半范畴化的数值特征对应。
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…
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