Quivers Of Type Research Articles

The Jordan-Hölder property of torsion-free classes in a quiver of type a

ABSTRACT We investigate the torsion-free classes satisfying the Jordan-Hölder property in the representation category of a quiver of type A. Explicitly, we use reflection functors and c-sortable elements to give a combinatorial criterion for determining such torsion-free classes.

Exceptional sequences and rooted labeled forests

We give a representation-theoretic bijection between rooted labeled forests with [Formula: see text] vertices and complete exceptional sequences for the quiver of type [Formula: see text] with straight orientation. The ascending and descending vertices in the forest correspond to relatively injective and relatively projective objects in the exceptional sequence. We conclude that every object in an exceptional sequence for linearly oriented [Formula: see text] is either relatively projective or relatively injective or both. We construct a natural action of the extended braid group on rooted labeled forests and show that it agrees with the known action of the braid group on complete exceptional sequences. We also describe the action of [Formula: see text], the Garside element of the braid group, on rooted labeled forests using representation theory and show how this relates to cluster theory.

On homological properties of the category of [formula omitted]-representations over a linear quiver of type [formula omitted

Let Q be a quiver of type An with linear orientation and rep(Q,F1) the category of representations of Q over the virtual field F1. It is proved that rep(Q,F1) has global dimension 2 whenever n≥3 and it is hereditary if n≤2. As a consequence, the Euler form 〈L,M〉=∑i=0∞dim⁡Exti(L,M) is well-defined. However, it does not descend to the Grothendieck group of rep(Q,F1). This yields negative answers to questions raised by Szczesny (2012) [4].

Presentations of Braid Groups of Type A Arising from (m+2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(m+2)$$\\end{document}-angulations of Regular Polygons

Coloured quiver mutation, introduced by Buan, A.B., Thomas, H (Adv. Math. 222(3), 971–995 2009), gives a combinatorial interpretation of tilting in higher cluster categories. In type A work of Baur, K., Marsh, B. (Trans. Am. Math. Soc. 360(11), 5789-5803 2008) shows that m-coloured quivers and m-coloured quiver mutations have a nice geometrical description, given in terms of (m+2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(m+2)$$\\end{document}-angulations of a regular polygon, and rotations of an m-diagonal. In this paper, using such correspondence, we describe presentations of braid groups of type A arising from coloured quivers of mutation type A.

Saw varieties

We introduce a new class of quiver varieties, called saw (quiver) varieties. These can be seen as a generalization of the three existing theories: the quiver gauge theory of monopoles, Kac's representation theory of Dynkin types, and Donaldson's theory on the based maps of P1. E.g., in our terminology, the handsaw (quiver) varieties and chainsaw varieties are finite and affine A type saw varieties respectively with the maximal symmetry breaking. These varieties are the monopole spaces and the caloron spaces in 3d gauge theory, respectively. We expand the theory over to the DE type quiver gauge theory.The saw varieties are constructed via a Hamiltonian formalism, so that the moment maps defining these varieties naturally arise. The main assertion of the paper is that the moment map μ defining a saw variety is a flat morphism for arbitrary dimension vector, if and only if the underlying graph is of finite ADE types. Furthermore under the assumption of maximal symmetry breaking or all the higher frame dimension vectors, the affine ADE types also attain the flatness property of μ. We give an algebro-geometric description of the momentum-zero scheme μ−1(0) and the saw variety defined as the Hamiltonian reduction.Various partition functions given rise to by the above mentioned theories are the byproducts of the geometric structure of the corresponding saw varieties.

Grothendieck lines in 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SQCD and the quantum K-theory of the Grassmannian

We revisit the 3d GLSM computation of the equivariant quantum K-theory ring of the complex Grassmannian from the perspective of line defects. The 3d GLSM onto X = Gr(Nc, nf) is a circle compactification of the 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetric gauge theory with gauge group UNck,k+lNc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{U}{\\left({N}_c\\right)}_{k,k+l{N}_c} $$\\end{document} and nf fundamental chiral multiplets, for any choice of the Chern-Simons levels (k, l) in the ‘geometric window’. For k=Nc−nf2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ k={N}_c-\\frac{n_f}{2} $$\\end{document} and l = −1, the twisted chiral ring generated by the half-BPS lines wrapping the circle has been previously identified with the quantum K-theory ring QKT(X). We identify new half-BPS line defects in the UV gauge theory, dubbed Grothendieck lines, which flow to the structure sheaves of the (equivariant) Schubert varieties of X. They are defined by coupling N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetric gauged quantum mechanics of quiver type to the 3d GLSM. We explicitly show that the 1d Witten index of the defect worldline reproduces the Chern characters for the Schubert classes, which are written in terms of double Grothendieck polynomials. This gives us a physical realisation of the Schubert-class basis for QKT(X). We then use 3d A-model techniques to explicitly compute QKT(X) as well as other K-theoretic enumerative invariants such as the topological metric. We also consider the 2d/0d limit of our 3d/1d construction, which gives us local defects in the 2d GLSM, the Schubert defects, that realise equivariant quantum cohomology classes.

Classification results for n-hereditary monomial algebras

We classify n-hereditary monomial algebras in three natural contexts: First, we give a classification of the n-hereditary truncated path algebras. We show that they are exactly the n-representation-finite Nakayama algebras classified by Vaso. Next, we classify partially the n-hereditary quadratic monomial algebras. In the case n=2, we prove that there are only two examples, provided that the preprojective algebra is a planar quiver with potential. The first one is a Nakayama algebra and the second one is obtained by mutating kA3⊗kkA3, where A3 is the Dynkin quiver of type A with bipartite orientation. In the case n≥3, we show that the only n-representation finite algebras are the n-representation-finite Nakayama algebras with quadratic relations.

On some Ringel-Hall polynomials associated to tame indecomposable modules

Let k be a finite field and Q an acyclic quiver of tame type (i.e. of type A˜m,D˜m,E˜6,E˜7,E˜8). Consider the path algebra kQ and the category of finite dimensional right modules mod-kQ. Using Schofield induction combined with some reduction tools, we determine all the Ringel-Hall polynomials associated to indecomposable modules of defect ranging from −2 to 2. In this way we obtain the full list of Ringel-Hall polynomials corresponding to indecomposables in the tame cases A˜m, D˜m and a partial list in the tame cases E˜6, E˜7, E˜8.

Ringel–Hall polynomials associated to a quiver of type $${\tilde{D}}_{4}$$

Ringel–Hall polynomials associated to a quiver of type $${\tilde{D}}_{4}$$

Multi-layer quivers and higher slice algebras

In this paper, we introduce multi-layer quivers and show how to construct an [Formula: see text]-slice algebra of infinite type from an [Formula: see text]-slice algebra of infinite type using the bound quivers. This leads to constructing [Formula: see text]-slice algebras of infinite type as matrix algebra and as tensor algebra of an [Formula: see text]-slice algebra and equivalences of their module categories as the module categories of diagram of some quiver of type [Formula: see text].

Approximation by interval-decomposables and interval resolutions of persistence modules

In topological data analysis, two-parameter persistence can be studied using the representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of type A. In a previous work, we defined interval approximations using restrictions to essential vertices of intervals together with Mobius inversion. In this work, we consider homological approximations using interval resolutions, and show that the interval resolution global dimension is finite for finite posets and that it is equal to the maximum of the interval dimensions of the Auslander-Reiten translates of the interval representations. In fact, for the latter equality, we obtained a general formula in the setting of finite-dimensional algebras and resolutions relative to a generator-cogenerator. Furthermore, in the commutative ladder case, by a suitable modification of our interval approximation, we provide a formula linking the two conceptions of approximation.

Massive flows in AdS6/CFT5

We study five-dimensional N=1 Superconformal Field Theories of the linear quiver type. These are deformed by a relevant operator, corresponding to a homogeneous mass term for certain matter fields. The free energy is calculated at arbitrary values of the mass parameter. After a careful regularisation procedure, the result can be put in correspondence with a calculation in the supergravity dual background. The F-theorem is verified for these flows, both in field theory and in supergravity. This letter presents some of the results in the companion paper [1].

Exchange graphs for mutation-finite non-integer quivers

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows us to define a geometric notion of seeds and thus to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers. The concept of finite type generalises naturally to mutation-finite non-integer quivers. We show that for all non-integer quivers of finite type there is a well-defined notion of an exchange graph, and this notion is consistent with the classical notion of exchange graph of integer mutation types coming from cluster algebras. In particular, exchange graphs of finite type quivers are finite. We also show that exchange graphs of rank 3 affine quivers are finite modulo the action of a finite-dimensional lattice (but unlike the integer case, the rank of the lattice is higher than 1 for non-integer quivers).

Red Sizes of Quivers

In this article, we will expand on the notions of maximal green and reddening sequences for quivers associated to cluster algebras. The existence of these sequences has been studied for a variety of applications related to Fomin and Zelevinsky's cluster algebras. Ahmad and Li considered a numerical measure of how close a quiver is to admitting a maximal green sequence called a red number. In this paper we generalized this notion to what we call unrestricted red numbers which are related to reddening sequences. In addition to establishing this more general framework we completely determine the red numbers and unrestricted red numbers for all finite mutation type quivers. Furthermore, we give conjectures on the possible values of red numbers and unrestricted red numbers in general.

Computation of extension spaces for the path algebra of type Ã(n - 1,1) using planar curves

Q is a quiver of type if its graph is of affine type and if its arrows have a certain orientation. We develop a bijection between the set of indecomposable kQ-modules whose dimension vectors are positive real roots of the root system associated to Q and a certain set of planar curves. We prove that the number of self-intersections of the curve which corresponds to the module M is equal to the dimension of . We also prove that, for many pairs of modules (M, N), the number of intersections between the corresponding two curves is equal to the dimension of , where C is the cluster category of kQ-mod.

On total stability conditions for Dynkin quivers

AbstractWe show that for a Dynkin quiver of type with a specific orientation, the path algebra has no slope function of the form that defines a total stability condition. This gives a counterexample to a conjecture of Reineke.

Continuous Quivers of Type A (IV)

This if the final paper in the series Continuous Quivers of Type A. In this part, we generalize existing geometric models of type A cluster structures for the new E-clusters introduced in part (III). We also introduce an isomorphism of cluster theories and a weak equivalence of cluster theories. Examples of both are given. We use these geometric models and isomorphisms of cluster theories to begin classifying continuous type A cluster theories. We also introduce a continuous generalization of mutation. This encompasses mutation and (infinite) sequences of mutation. Then we link continuous mutation to our earlier geometric models. Finally, we introduce the space of mutations which generalizes the exchange graph of a cluster structure, and show that paths in this space are continuous mutations.

CANONICAL DECOMPOSITION AND QUIVER REPRESENTATIONS OF TYPE OVER FINITE FIELDS

AbstractLet Q be a quiver of type $\tilde {A}_n$ . Let $\alpha =\alpha _1+\alpha _2+\cdots +\alpha _s$ be the canonical decomposition. For the polynomials $M_Q(\alpha ,q)$ that count the number of isoclasses of representations of Q over ${\mathbb F}_q$ with dimension vector $\alpha $ , we obtain a precise relation between the degree of $M_Q(\alpha ,q)$ and that of $\prod _{i=1}^{s} M_Q(\alpha _i,q)$ for an arbitrary dimension vector $\alpha $ .

Tangent Spaces of Orbit Closures for Representations of Dynkin Quivers of Type mathbb {D}

Let Bbbk be an algebraically closed field, Q a finite quiver, and denote by textup {rep}_{Q}^{mathbf {d}} the affine Bbbk -scheme of representations of Q with a fixed dimension vector d. Given a representation M of Q with dimension vector d, the set {mathcal {O}}_{M} of points in Bbbk isomorphic as representations to M is an orbit under an action on textup {rep}^{mathbf {d}}_{Q}Bbbk of a product of general linear groups. The orbit {mathcal {O}}_{M} and its Zariski closure overline {mathcal {O}}_{M}, considered as reduced subschemes of textup {rep}_{Q}^{{mathbf {d}}}, are contained in an affine scheme {mathcal {C}}_{M} defined by suitable rank conditions associated to M. For all Dynkin and extended Dynkin quivers, the sets of points of overline {{mathcal {O}}}_{M} and {mathcal {C}}_{M} coincide, or equivalently, overline {{mathcal {O}}}_{M} is the reduced scheme associated to {mathcal {C}}_{M}. Moreover, overline {mathcal {O}}_{M}={mathcal {C}}_{M} provided Q is a Dynkin quiver of type {mathbb {A}}, and this equality is a conjecture for the remaining Dynkin quivers (of type mathbb {D} and {mathbb {E}}). Let Q be a Dynkin quiver of type mathbb {D} and M a finite dimensional representation of Q. We show that the equality T_{N}overline {mathcal {O}}_{M}=T_{N}{mathcal {C}}_{M} of Zariski tangent spaces holds for any closed point N of overline {mathcal {O}}_{M}. As a consequence, we describe the tangent spaces to overline {mathcal {O}}_{M} in representation theoretic terms.

Tilting modules arising from knot invariants

We construct tilting modules over Jacobian algebras arising from knots. To a two-bridge knot L [ a 1 , … , a n ] , we associate a quiver Q with potential and its Jacobian algebra A . We construct a family of canonical indecomposable A -modules M ( i ) , each supported on a different specific subquiver Q ( i ) of Q . Each of the M ( i ) is expected to parametrize the Jones polynomial of the knot. We study the direct sum M = ⊕ i M ( i ) of these indecomposables inside the module category of A as well as in the cluster category. In this paper we consider the special case where the two-bridge knot is given by two parameters a 1 , a 2 . We show that the module M is rigid and τ -rigid, and we construct a completion of M to a tilting (and τ -tilting) A -module T . We show that the endomorphism algebra End A T of T is isomorphic to A , and that the mapping T ↦ A [ 1 ] induces a cluster automorphism of the cluster algebra A ( Q ) . This automorphism is of order two. Moreover, we give a mutation sequence that realizes the cluster automorphism. In particular, we show that the quiver Q is mutation equivalent to an acyclic quiver of type T p , q , r (a tree with three branches). This quiver is of finite type if ( a 1 , a 2 ) = ( a 1 , 2 ) , ( 1 , a 2 ) , or ( 2 , 3 ) , it is tame for ( a 1 , a 2 ) = ( 2 , 4 ) or ( 3 , 3 ) , and wild otherwise.