Quiver Representations Research Articles

Laumon parahoric local models via quiver Grassmannians

Local models of Shimura varieties in type A can be realized inside products of Grassmannians via certain linear algebraic conditions. Laumon suggested a generalization which can be identified with a family over a line whose general fibers are quiver Grassmannians for the loop quiver and the special fiber is a certain quiver Grassmannian for the cyclic quiver. The whole family sits inside the Gaitsgory central degeneration of the affine Grassmannians. We study the properties of the special fibers of the (complex) Laumon local models for arbitrary parahoric subgroups in type A using the machinery of quiver representations. We describe the irreducible components and the natural strata with respect to the group action for the quiver Grassmannians in question. We also construct a cellular decomposition and provide an explicit description for the corresponding poset of cells. Finally, we study the properties of the desingularizations of the irreducible components and show that the desingularization construction is compatible with the natural projections between the parahoric subgroups.

Generalized juggling patterns, quiver Grassmannians and affine flag varieties

The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmannians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, in particular, to describe explicitly embeddings into affine flag varieties which allow us to realize our quiver Grassmannians as a union of Schubert varieties therein.

Symplectic Grassmannians and cyclic quivers

AbstractThe goal of this paper is to extend the quiver Grassmannian description of certain degenerations of Grassmann varieties to the symplectic case. We introduce a symplectic version of quiver Grassmannians studied in our previous papers and prove a number of results on these projective algebraic varieties. First, we construct a cellular decomposition of the symplectic quiver Grassmannians in question and develop combinatorics needed to compute Euler characteristics and Poincaré polynomials. Second, we show that the number of irreducible components of our varieties coincides with the Euler characteristic of the classical symplectic Grassmannians. Third, we describe the automorphism groups of the underlying symplectic quiver representations and show that the cells are the orbits of this group. Lastly, we provide an embedding into the affine flag varieties for the affine symplectic group.

Rigid integral representations of quivers over arbitrary commutative rings

Rigid integral representations of quivers over arbitrary commutative rings

A functorial approach to monomorphism categories II: Indecomposables

AbstractWe investigate the (separated) monomorphism category of a quiver over an Artin algebra . We show that there exists an epivalence (called representation equivalence in the terminology of Auslander) from to , where is the category of finitely generated ‐modules and and denote the respective injectively stable categories. Furthermore, if has at least one arrow, then we show that this is an equivalence if and only if is hereditary. In general, the epivalence induces a bijection between indecomposable objects in and noninjective indecomposable objects in , and we show that the generalized Mimo‐construction, an explicit minimal right approximation into , gives an inverse to this bijection. We apply these results to describe the indecomposables in the monomorphism category of radical square zero Nakayama algebras, and to give a bijection between the indecomposables in the monomorphism category of two artinian uniserial rings of Loewy length 3 with the same residue field. The main tool to prove these results is the language of a free monad of an exact endofunctor on an arbitrary abelian category. This allows us to avoid the technical combinatorics arising from quiver representations. The setup also specializes to more general settings, such as representations of modulations. In particular, we obtain new results on the singularity category of the algebras that were introduced by Geiss, Leclerc, and Schröer in order to extend their results relating cluster algebras and Lusztig's semicanonical basis to symmetrizable Cartan matrices. We also recover results on the algebras that were introduced by Lu and Wang to realize groups via semiderived Hall algebras.

A classification of polyharmonic Maaß forms via quiver representations

We give a classification of the Harish-Chandra modules generated by the pullback to SL2(R) of polyharmonic Maaß forms for congruence subgroups of SL2(Z) with exponential growth allowed at the cusps. This extends results of Bringmann–Kudla in the harmonic case. While in the harmonic setting there are nine cases, our classification comprises ten; A new case arises in weights k>1. To obtain the classification we introduce quiver representations into the topic and show that those associated with polyharmonic Maaß forms are cyclic, indecomposable representations of the two-cyclic or the Gelfand quiver. A classification of these transfers to a classification of polyharmonic weak Maaß forms. To realize all possible cases of Harish-Chandra modules we develop a theory of weight shifts for Taylor coefficients of vector-valued spectral families. We provide a comprehensive computer implementation of this theory, which allows us to provide explicit examples.

Reductive quotients of klt singularities

We prove that the quotient of a klt type singularity by a reductive group is of klt type in characteristic 0. In particular, given a klt variety X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document} endowed with the action of a reductive group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G$\\end{document} and admitting a quasi-projective good quotient X→X//G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X\\rightarrow X/\\!/G$\\end{document}, we can find a boundary B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$B$\\end{document} on X//G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/G$\\end{document} so that the pair (X//G,B)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(X/\\!/G,B)$\\end{document} is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G$\\end{document}-manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n$\\end{document}-dimensional K-polystable smooth Fano varieties of volume v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$v$\\end{document} has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.

Relative tilting theory in abelian categories I: Auslander–Buchweitz–Reiten approximations theory in subcategories and cotorsion-like pairs

In this paper, we introduce a special kind of relative (co)resolutions associated with a pair of classes of objects in an abelian category C.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}.$$\\end{document} We will see that, by studying this relative (co)resolutions, we get a possible generalization of a part of the Auslander–Buchweitz approximation theory that is useful for developing n-X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-tilting theory in Argudin Monroy and Mendoza Hernández (relative tilting theory in abelian categories II: n-X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document} tilting theory. arXiv:2112.14873, 2021). With this goal, new concepts as X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-complete and X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-hereditary pairs are introduced as a generalization of complete and hereditary cotorsion pairs. These pairs appear in a natural way in the study of the category of representations of a quiver in an abelian category (Argudin Monroy and Mendoza Hernández in categories of quiver representations and relative cotorsion pairs. arXiv:2311.12774v1, 2023). Our main results will include an existence theorem for relative approximations, among other results related with closure properties of relative (co)resolution classes and relative homological dimensions which are essential in the development of n-X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-tilting theory in Argudin Monroy and Mendoza Hernández (2021).

Bigraded Betti numbers and generalized persistence diagrams

Commutative diagrams of vector spaces and linear maps over Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}^2$$\\end{document} are objects of interest in topological data analysis (TDA) where this type of diagrams are called 2-parameter persistence modules. Given that quiver representation theory tells us that such diagrams are of wild type, studying informative invariants of a 2-parameter persistence module M is of central importance in TDA. One of such invariants is the generalized rank invariant, recently introduced by Kim and Mémoli. Via the Möbius inversion of the generalized rank invariant of M, we obtain a collection of connected subsets I⊂Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I\\subset \\mathbb {Z}^2$$\\end{document} with signed multiplicities. This collection generalizes the well known notion of persistence barcode of a persistence module over R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document} from TDA. In this paper we show that the bigraded Betti numbers of M, a classical algebraic invariant of M, are obtained by counting the corner points of these subsets Is. Along the way, we verify that an invariant of 2-parameter persistence modules called the interval decomposable approximation (introduced by Asashiba et al.) also encodes the bigraded Betti numbers in a similar fashion. We also show that the aforementioned results are optimal in the sense that they cannot be extended to d-parameter persistence modules for d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 3$$\\end{document}.

On Chow Rings of Quiver Moduli

Abstract We describe the point class and Todd class in the Chow ring of a moduli space of quiver representations, building on a result of Ellingsrud–Strømme. This, together with the presentation of the Chow ring by the second author, makes it possible to compute integrals on quiver moduli. To do so, we construct a canonical morphism of universal representations in great generality, and along the way point out its relation to the Kodaira–Spencer morphism. We illustrate the results by computing some invariants of some “small” Kronecker moduli spaces. We also prove that the first non-trivial (6-dimensional) Kronecker moduli space is isomorphic to the zero locus of a general section of $\mathcal{Q}^{\vee }(1)$ on $\textrm{Gr}(2,8)$.

A deterministic algorithm for Harder–Narasimhan filtrations for representations of acyclic quivers

A deterministic algorithm for Harder–Narasimhan filtrations for representations of acyclic quivers

Tensor product of representations of quivers

In this article, we define the tensor product V⊗W of a representation V of a quiver Q with a representation W of an another quiver Q′, and show that the representation V⊗W is semistable if V and W are semistable. We give a relation between the universal representations on the fine moduli spaces N1,N2 and N3 of representations of Q,Q′ and Q⊗Q′ respectively over arbitrary algebraically closed fields. We further describe a relation between the natural line bundles on these moduli spaces when the base is the field of complex numbers. We then prove that the internal product Q̃⊗Q′̃ of covering quivers is a sub-quiver of the covering quiver Q⊗Q′˜. We deduce the relation between stability of the representations V⊗W˜ and Ṽ⊗W̃, where Ṽ denotes the lift of the representation V of Q to the covering quiver Q̃. We also lift the relation between the natural line bundles on the product of moduli spaces N1̃×N2̃.

Locally free representations of quivers over commutative Frobenius algebras

Locally free representations of quivers over commutative Frobenius algebras

COEFFICIENT QUIVERS, ${\mathbb {F}}_1$ -REPRESENTATIONS, AND EULER CHARACTERISTICS OF QUIVER GRASSMANNIANS

AbstractA quiver representation assigns a vector space to each vertex, and a linear map to each arrow of a quiver. When one considers the category $\mathrm {Vect}(\mathbb {F}_1)$ of vector spaces “over $\mathbb {F}_1$ ” (the field with one element), one obtains $\mathbb {F}_1$ -representations of a quiver. In this paper, we study representations of a quiver over the field with one element in connection to coefficient quivers. To be precise, we prove that the category $\mathrm {Rep}(Q,\mathbb {F}_1)$ is equivalent to the (suitably defined) category of coefficient quivers over Q. This provides a conceptual way to see Euler characteristics of a class of quiver Grassmannians as the number of “ $\mathbb {F}_1$ -rational points” of quiver Grassmannians. We generalize techniques originally developed for string and band modules to compute the Euler characteristics of quiver Grassmannians associated with $\mathbb {F}_1$ -representations. These techniques apply to a large class of $\mathbb {F}_1$ -representations, which we call the $\mathbb {F}_1$ -representations with finite nice length: we prove sufficient conditions for an $\mathbb {F}_1$ -representation to have finite nice length, and classify such representations for certain families of quivers. Finally, we explore the Hall algebras associated with $\mathbb {F}_1$ -representations of quivers. We answer the question of how a change in orientation affects the Hall algebra of nilpotent $\mathbb {F}_1$ -representations of a quiver with bounded representation type. We also discuss Hall algebras associated with representations with finite nice length, and compute them for certain families of quivers.

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.

Artificial Neural Networks Using Quiver Representations of Finite Cyclic Groups

Artificial Neural Networks Using Quiver Representations of Finite Cyclic Groups

Jet schemes, quantum dilogarithm and Feigin-Stoyanovsky's principal subspaces

We analyze the structure of Feigin-Stoyanovsky's principal subspaces of affine Lie algebra from the jet algebra viewpoint. For type A level one principal subspaces, we show that their shifted multi-graded Hilbert series can be expressed either using the quantum dilogarithm or as certain generating functions “counting” finite-dimensional representations of A-type quivers. This notably results in novel fermionic character formulas for these principal subspaces. Moreover, our result implies that all level one principal subspaces of type A are “classically free” as vertex algebras.We also analyze infinite jet algebras associated to principal subspaces of affine vertex algebras L1(so5), L1(so8) and L1(g2). We derive a new character formula for the principal subspace of L1(so5), proving that it is classically free, and present evidence that the principal subspaces of L1(so8) and of L1(g2) are also classically free.

Decidability of flow equivalence and isomorphism problems for graph C*-algebras and quiver representations

We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z (the integers). Consequently, results of Eilers, Restorff, Ruiz and Sørensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups).

Representations of free products of semisimple algebras via quivers

We use quiver representation theory and King's notion of stable representations to study the (finite dimensional) representations of free products of semi-simple associative K-algebras over an algebraically closed field K. We obtain numerical criteria giving necessary conditions for a module over such a free product to be simple and prove that these conditions are sufficient for modules in general position. We also prove that modules in general position are always semisimple, although non-semisimple modules always exist when the algebra is infinite dimensional. With essentially a single exception, all module categories are strictly wild; moreover these categories admit no bounds on the dimensions of simple modules. We use quiver moduli spaces to derive a closed formula for the number of parameters needed to describe all simple modules in a given dimension. As a special case, we apply our results to free products of finite groups, and recover the representation theory of the projective modular group PSL2(Z) (strictly wild) and of the infinite dihedral group (tame).

Tensor stable moduli stacks and refined representations of quivers

AbstractIn this paper, we look at the problem of modular realisations of derived equivalences, and more generally, the problem of recovering a Deligne–Mumford stack and a bundle on it, via some moduli problem (on or ). The key issue is, how does one incorporate some of the monoidal structure of into the moduli problem. To this end, we introduce a new moduli stack, the tensor stable moduli stack which generalises the notion of the Serre‐stable moduli stack. We then show how it can be used both for stack recovery and the modular realisation problem for derived equivalences. We also study the moduli of refined representations and how they address these problems. Finally, we relate the two approaches when is a tilting bundle which is a direct sum of line bundles.