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 endomorphism ring of a given real root representation.
Acyclic cluster algebras have an interpretation in terms of tilting objects in a Calabi-Yau category defined by some hereditary algebra. For a given quiver $Q$ it is thus desirable to decide if the cluster algebra defined by $Q$ is acyclic. We call $Q$ cluster-acyclic in this case, otherwise cluster-cyclic. In this note we classify the cluster-cyclic quivers with three vertices using a Diophantine equation studied by Markov.
In this paper we completely describe maximal green sequences (MGS) of acyclic quivers with multiple edges in terms of maximal green sequences of their multiple edge-free (ME-free) versions. In particular we establish that any MGS of a quiver must be an ME-free MGS of its ME-free version. For quivers with oriented cycles our results for acyclic quivers do not apply. However a weaker result is available under certain conditions.
We describe the class of quiver settings with one dimensional vertices whose semi-simple representations are parametrized by a complete intersection variety. We show that these quivers can be reduced to a one vertex quiver with some combinatorial reduction steps. We also show that this class consists of the quivers from which we can not obtain two specific non complete intersection quivers via contracting strongly connected components and deleting subquivers. The class of coregular quiver settings with arbitrary dimension vector, that has been described by an earlier result via reduction steps…
Let Delta be an Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of Delta are normal and Cohen--Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.
A maximal green sequence introduced by B. Keller is a certain sequence of quiver mutations at green vertices. T. Brüstle, G. Dupont and M. Pérotin showed that for an acyclic quiver, maximal green sequences are realized as maximal paths in the Hasse quiver of the poset of support tilting modules. They also considered possible lengths of maximal green sequences. In this paper, we calculate possible lengths of maximal green sequences for a quiver of type $A$ or of type $\tilde{A}_{n,1}$ by using theory of tilting mutation.
In general, the existence of a maximal green sequence is not mutation invariant. In this paper we show that it is in fact mutation invariant for cluster quivers of finite mutation type. In particular, we show that a mutation finite cluster quiver has a maximal green sequence unless it arises from a once-punctured closed marked surface, or one of the two quivers in the mutation class of X7. We develop a procedure to explicitly find maximal green sequences for cluster quivers associated to arbitrary triangulations of closed marked surfaces with at least two punctures. As a corollary, it follows …
An upper bound on degrees of elements of a minimal generating system for invariants of quivers of dimension (2,...,2) is established over a field of arbitrary characteristic and its precision is estimated. The proof is based on the reduction to the problem of description of maximal paths satisfying certain condition.
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$, and $M$ be a partial tilting $A$-module. We prove that the Bongartz $τ$-tilting complement of $M$ coincides with its Bongartz complement, and then we give a new proof of that every almost complete tilting $A$-module has at most two complements. Let $A=kQ$ be a path algebra. We prove that the support $τ$-tilting quiver $\overrightarrow{Q}({\rm s}τ$-${\rm tilt} A)$ of $A$ is connected. As an application, we investigate the conjecture of Happel and Unger in [9] which claims that each connected component of the tilting…
We prove quantum dilogarithm identities for $n$-cycle quivers. By the combinatorial approach of Keller, each side of our identity determines a maximal green sequence of quiver mutations. Thus we interpret our identities as factorizations of the refined Donaldson--Thomas invariant for the quiver with potential. Finally, we conjecture an upper bound on the possible lengths of maximal green sequences.
Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. We take the categorical notion and introduce maximal green sequences for hearts of bounded t-structures of triangulated categories that can be tilted indefinitely. We study the case where the heart is the category of modules over the preprojective algebra of a quiver without loops. The combinatorical counterpart of maximal green sequences for Dynkin quivers are maximal chains in the Hasse quiver of basic support τ-tilting modules. We show that…
Given a framed quiver, i.e. one with a frozen vertex associated to each mutable vertex, there is a concept of green mutation, as introduced by Keller. Maximal sequences of such mutations, known as maximal green sequences, are important in representation theory and physics as they have numerous applications, including the computations of spectrums of BPS states, Donaldson-Thomas invariants, tilting of hearts in the derived category, and quantum dilogarithm identities. In this paper, we study such sequences and construct a maximal green sequence for every quiver mutation-equivalent to an orienta…
Maximal green sequences are particular sequences of quiver mutations appearing in the context of quantum dilogarithm identities and supersymmetric gauge theory. Interpreting maximal green sequences as paths in various natural posets arising in representation theory, we prove the finiteness of the number of maximal green sequences for cluster finite quivers, affine quivers and acyclic quivers with at most three vertices. We also give results concerning the possible numbers and lengths of these maximal green sequences.
We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras.
This paper presents a geometric model of the Auslander-Reiten quiver of a type A quiver together with a stability function for which all indecomposable modules are stable. We also introduce a new Catalan object which we call a maximal almost rigid representation. We show that its endomorphism algebra is a tilted algebra of type A. We define a partial order on the set of maximal almost rigid representations and use our new geometric model to show that this partial order is a Cambrian lattice.
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.
Let $Q$ be an $n$-Kronecker quiver, i.e., a quiver with two vertices, labeled by 1 and 2, and $n$ arrows from 2 to 1. We show that there are infinitely many Gabriel-Roiter measures admitting no direct predecessors.
We use semi-invariant pictures to prove two conjectures about maximal green sequences. First: if $Q$ is any acyclic valued quiver with an arrow $j\to i$ of infinite type then any maximal green sequence for $Q$ must mutate at $i$ before mutating at $j$. Second: for any quiver $Q'$ obtained by mutating an acyclic valued quiver $Q$ of tame type, there are only finitely many maximal green sequences for $Q'$. Both statements follow from the Rotation Lemma for reddening sequences and this in turn follows from the Mutation Formula for the semi-invariant picture for $Q$.
We use combinatorics of quivers and the corresponding surfaces to study maximal green sequences of minimal length for quivers of type $\mathbb{A}$. We prove that such sequences have length $n+t$, where $n$ is the number of vertices and $t$ is the number of 3-cycles in the quiver. Moreover, we develop a procedure that yields these minimal length maximal green sequences.
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…
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 not 3. We study how this quiver with potential changes when $T$ is mutated. We also provide a derived equivalence classification for the endomorphism algebras of maximal rigid objects.
We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem. In the special case of Dynkin quivers with n vertices this gives the fundamental interrelationship between supports of the semi-invariants and the Tilting Triangulation of the (n-1)-sphere.
We 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.
We introduce a new method for producing both maximal green and reddening sequences of quivers. The method, called component preserving mutations, generalizes the notion of direct sums of quivers and can be used as a tool to both recover known reddening sequences as well as find reddening sequences that were previously unknown. We use the method to produce and recover maximal green sequences for many bipartite recurrent quivers that show up in the study of periodicity of $T$-systems and $Y$-systems. Additionally, we show how our method relates to the dominance phenomenon recently considered by …
To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. The construction of reflection functors for quiver varieties are generalized to our case, in which these relate to simple reflections in the Weyl group of some symmetrizable, possibly non-symmetric Kac-Moody algebra. The moduli spaces of meromorphic connections on the rank 2 trivial bundle over the Riemann…
We prove that the frieze sequences of cluster variables associated with the vertices of an affine quiver satisfy linear recurrence relations. In particular, we obtain a proof of a recent conjecture by Assem-Reutenauer-Smith.
In this paper we show that for a simply-laced root system a choice of $C$ gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to $C$-orbits in $R$; moreover, it gives an identification of $R$ with a certain subset $Ihat$ of $I x Z_{2h}$, where $h$ is the Coxeter number. The set $Ihat$ has a natural quiver structure; we call it the periodic Auslander-Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram $I$: roots are vertices of $Ihat$, and the root lattice and the inner product admit an …
In this paper, we study the maximal length of maximal green sequences for quivers of type $\widetilde{\mathbf{D}}$ and $\widetilde{\mathbf{E}}$ by using the theory of tilting mutation. We show that the maximal length does not depend on the choice of the orientation, and determine it explicitly. Moreover, we give a program which counts all maximal green sequences by length for a given Dynkin/extended Dynkin quiver.
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.
We consider the base change action on real or complex representation spaces of quivers and the associated momentum map for a maximal compact subgroup of the base change group, as introduced by A. King. We give an explicit description of the momentum map image in terms of recursively defined inequalities on eigenvalues of Hermitian operators. Moreover, we characterize when the momentum map image is maximal possible, respectively of positive volume.
This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the gr…
Centre for Theoretical Physics, Queen Mary University of London（伦敦玛丽女王大学理论物理中心）；Department of Mathematics, University of Connecticut（康奈尔大学数学系）；Korea Institute for Advanced Study（韩国高等研究院）
It is well known that any triangulation of a marked surface produces a quiver. In this paper we will provide a triangulation for orientable surfaces of genus $n$ with an arbitrary number interior marked points (called punctures) whose corresponding quiver has a maximal green sequence.
We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cell…
This paper studies connections between the preprojective modules over the path algebra of a finite connected quiver without oriented cycles, the (+)-admissible sequences of vertices, and the Weyl group. For each preprojective module, there exists a unique up to a certain equivalence shortest (+)-admissible sequence annihilating the module. A (+)-admissible sequence is the shortest sequence annihilating some preprojective module if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. These statements have the following app…
Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. It is an open problem to determine what lengths are achieved by the maximal green sequences of a quiver. We combine the combinatorics of surface triangulations and the basics of scattering diagrams to address this problem. Our main result is a formula for the length of minimal length maximal green sequences of quivers defined by triangulations of an annulus or a punctured disk.
Let $Λ$ be a cluster-tilted algebra of finite type over an algebraically closed field and $B$ be one of the associated tilted algebras. We show that the $B$-modules, ordered form right to left in the Auslander-Reiten quiver of $Λ$ form a maximal forward hom-orthogonal sequence of $Λ$-modules whose dimension vectors form the $c$-vectors of a maximal green sequence for $Λ$. Thus we give a proof of Igusa-Todorov's conjecture.
Constraints are given on the depth of diagonal subalgebras in generalized triangular matrix algebras. The depth of the top subalgebra B = A /rad A in a finite, connected, acyclic quiver algebra A over an algebraically closed field K is then computed. Also the depth of the primary arrow subalgebra 1K + rad A = B in A is obtained. The two types of subalgebras have depths 3 and 4 respectively, independent of the number of vertices. An upper bound on depth is obtained for the quotient of a subalgebra pair.
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…
Let $\Aa_t$ be the directed quiver of type $\Aa$ with $t$ vertices. For each dimension vector $d$ there is a dense orbit in the corresponding representation space. The principal aim of this note is to use just rank conditions to define the irreducible components in the complement of the dense orbit. Then we compare this result with already existing ones by Knight and Zelevinsky, and by Ringel. Moreover, we compare with the fan associated to the quiver $\Aa$ and derive a new formula for the number of orbits using nilpotent classes. In the complement of the dense orbit we determine the irreducib…
The spectral radius of matrix, also known as Frobenius-Perron dimension, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, find a way to calculate the Frobenius-Perron dimension of these algebras when they satisfy the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver …
We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. By using $d-$cluster categories which are defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a $d-$compatibility degree $(-||-)$ on any pair of ``colored'' almost positive real Schur roots which generalizes previous definitions on the non-colored case, and call two such roots compatible provided the $d-$compatibility degree of them is zero. Associated to the root system $Φ$ corresponding to the va…
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.
Let G be a Lie group and Q a quiver with relations. In this paper, we define G-valued representations of Q which directly generalize G-valued representations of finitely generated groups. Although as G-spaces, the G-valued quiver representations are more general than G-valued representations of finitely generated groups, we show by collapsing arrows that their quotient spaces are equivalent. We then establish a general criterion for the moduli of G-valued quiver representations with relations to admit a strong deformation retraction to a compact quotient by pinching vertices on the quiver. Thi…
Let $Γ$ be a quiver on n vertices $v_1, v_2, ..., v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $α\in \N^n$. Hua gave a formula for $A_Γ(α, q)$, the number of isomorphism classes of absolutely indecomposable representations of $Γ$ over the finite field $\F_q$ with dimension vector $α$. Kac showed that $A_Γ(\bmα, q)$ is a polynomial in q with integer coefficients. Using Hua's formula, we show that for each non-negative integer s, the s-th derivative of $A_Γ(α,q)$ with respect to q, when evaluated at q = 1, is a polynomial in the variables $g_{ij}$, and we compute the highest degree …
The 3-Kronecker quiver has two vertices, namely a sink and a source, and 3 arrows. A regular representation of a representation-infinite quiver such as the 3-Kronecker quiver is said to be elementary provided it is non-zero and not a proper extension of two regular representations. Of course, any regular representation has a filtration whose factors are elementary, thus the elementary representations may be considered as the building blocks for obtaining all the regular representations. We are going to determine the elementary $3$-Kronecker modules. It turns out that all the elementary modules…
This paper deals with the representation theory of a locally finite quiver in which the number of paths between any two given vertices is finite. We first study some properties of the finitely presented or co-presented representations, and then construct in the category of locally finite dimensional representations some almost split sequences which start with a finitely co-presented representation and end with a finitely presented representation. Furthermore, we obtain a general description of the shapes of the Auslander-Reiten components of the category of finitely presented representations a…
In a recent paper by K.-H. Lee, K. Lee and M. Mills, a mutation of reflections in the universal Coxeter group is defined in association with a mutation of a quiver. A matrix representation of these reflections is determined by a linear ordering on the set of vertices of the quiver. It was conjectured that there exists an ordering (called a pseudo-acyclic ordering in this paper) such that whenever two mutation sequences of a quiver lead to the same labeled seed, the representations of the associated reflections also coincide. In this paper, we prove this conjecture for every quiver mutation-equ…
Gabriel's Theorem, and the work of Bernstein, Gelfand and Ponomarev established a connection between the theory of quiver representations and the theory of simple Lie algebras. Lie superalgebras have been studied from many perspectives, and many results about Lie algebras have analogues for Lie superalgebras. In this paper, the notion of a super-representation of a quiver is introduced, as well as the notion of reflection functors for odd roots. These ideas are then used to give a categorical construction of the root system A(n,m) by establishing a version of Gabriel's Theorem and modifying th…
Let A be a basic connected finite dimensional algebra over an algebraically closed field k and with ordinary quiver Q without oriented cycle. To any presentation of A by quiver and admissible relations, Martinez-Villa and de La Pena have associated the fundamental group of the presentation. Assem and de La Pena have constructed an injective mapping from the additive characters of this fundamental group (with values in the ground field) to the first Hochschild cohomology group HH^1(A). We study the image of these mappings associated to the different presentations of A in terms of diagonalizable…
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…
