Indecomposable Representations Research Articles

Special function models of indecomposable sl(2) representations: the Laguerre case

In this paper, we point out connections between certain types of indecomposable representations of sl(2) and generalizations of well-known orthogonal polynomials. Those representations take the form of infinite dimensional chains of weight or generalised weight spaces, for which the Cartan generator acts in a diagonal way or via Jordan blocks. The other generators of the Lie algebras sl(2) act as raising and lowering operators but are now allowed to relate the different chains as well. In addition, we construct generating functions, we calculate the action of the Casimir invariant and present relations to systems of non-homogeneous second-order coupled differential equations. We present different properties as higher-order linear differential equations for building blocks taking the form of one variable polynomials. We also present insight into the zeroes and recurrence relations.

Carroll fermions

Using carefully chosen projections, we consider different Carroll limits of relativistic Dirac fermions in any spacetime dimensions. These limits define Carroll fermions of two types: electric and magnetic. The latter type transforms as a reducible but indecomposable representation of the Carroll group. We also build action principles for all Carroll fermions we introduce; in particular, in even dimensions we provide an action principle for a minimal magnetic Carroll fermion, having the same number of components as a Dirac spinor. We then explore the coupling of these fermions to magnetic Carroll gravity in both its first-order and second-order formulations.

G-semisimple algebras

Let Λ be an Artin algebra and mod-(Gprj_-Λ) the category of finitely presented functors over the stable category Gprj_-Λ of finitely generated Gorenstein projective Λ-modules. This paper deals with those algebras Λ in which mod-(Gprj_-Λ) is a semisimple abelian category, and we call G-semisimple algebras. We study some basic properties of such algebras. In particular, it will be observed that the class of G-semisimple algebras contains important classes of algebras, including gentle algebras and more generally quadratic monomial algebras. Next, we construct an epivalence (called representation equivalence in the terminology of Auslander), i.e. a full and dense functor that reflects isomorphisms, from the stable category of Gorenstein projective representations Gprj_(Q,Λ) of a finite acyclic quiver Q to the category of representations rep(Q,Gprj_-Λ) over Gprj_-Λ, provided Λ is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra ΛQ of the G-semisimple algebra Λ is CM-finite if and only if Q is Dynkin. In the last part, we provide a complete classification of indecomposable Gorenstein projective representations within Gprj(An,Λ) of the linear quiver An over a G-semisimple algebra Λ. We also determine almost split sequences in Gprj(An,Λ) with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver Gprj(An,Λ).

On the Product Functor on Inner forms of the General Linear Group Over A Non-Archimedean Local Field

AbstractLet $$G_n$$ G n be an inner form of a general linear group over a non-Archimedean local field. We fix an arbitrary irreducible representation $$\sigma $$ σ of $$G_n$$ G n . Building on the work of Lapid-Mínguez on the irreducibility of parabolic inductions, we show how to define a full subcategory of the category of smooth representations of some $$G_m$$ G m , on which the parabolic induction functor $$\tau \mapsto \tau \times \sigma $$ τ ↦ τ × σ is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative.

Locally free Caldero–Chapoton functions via reflections

We study the reflections of locally free Caldero–Chapoton functions associated to representations of Geiß–Leclerc–Schröer’s quivers with relations for symmetrizable Cartan matrices. We prove that for rank 2 cluster algebras, non-initial cluster variables are expressed as locally free Caldero–Chapoton functions of locally free indecomposable rigid representations. Our method gives rise to a new proof of the locally free Caldero–Chapoton formulas obtained by Geiß–Leclerc–Schröer in Dynkin cases. For general acyclic skew-symmetrizable cluster algebras, we prove the formula for any non-initial cluster variable obtained by almost sink and source mutations.

Decompositions of Infinite-Dimensional A∞,∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_{\\infty , \\infty }$$\\end{document} Quiver Representations

Gabriel’s Theorem states that the quivers which have finitely many isomorphism classes of indecomposable representations are exactly those with underlying graph one of the ADE Dynkin diagrams and that the indecomposables are in bijection with the positive roots of this graph. When the underlying graph is An\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{A_n}$$\\end{document}, these indecomposable representations are thin (either 0 or 1 dimensional at every vertex) and in bijection with the connected subquivers. Using linear algebraic methods we show that every (possibly infinite-dimensional) representation of a quiver with underlying graph A∞,∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{A_{\\infty , \\infty }}$$\\end{document} is infinite Krull-Schmidt, i.e. a direct sum of indecomposables, as long as the arrows in the quiver eventually point outward. We furthermore prove that these indecomposable are again thin and in bijection with both the connected subquivers and the limits of the positive roots of A∞,∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{A_{\\infty , \\infty }}$$\\end{document} with respect to a certain uniform topology on the root space. Finally we give an example of an A∞,∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{A_{\\infty , \\infty }}$$\\end{document} quiver which is not infinite Krull-Schmidt and hence necessarily is not eventually-outward.

Invariant differential operators for noncompact Lie groups: The Sp(n,1) case

In this paper, we continue the project of systematic construction of invariant differential operators (IDOs) on the example of the noncompact algebras [Formula: see text]. Our choice of these algebras is motivated by the fact that they belong to a narrow class of algebras, which are of split rank one, of which class the other cases were studied, some long time ago. We concentrate on the case [Formula: see text]. We give the main multiplets and the main reduced multiplets of indecomposable elementary representations (ER) for, including the necessary data for all relevant invariant differential operators. We also present explicit expressions for the singular vectors and the intertwining differential operators.

Einstein Manifold of SO(p,q) with a New Parabolic Subgroup and Invariant Differential Operators

In the present paper we continue the project of systematic explicit construction of invariant differential operators. On the example of the non-compact group SO(p, 9 − p) (for p = 5, 6) we give the classification multiplets of indecomposable elementary representations induced from a new (relative to earlier considerations) choice of parabolic subgroup MAN so that the factor-group G/M is an Einstein manifold. This classification includes the data for the relevant invariant differential operators.

Einstein Manifold of SO(p, 9 – p), Its Corresponding Parabolic Subgroup, and Invariant Differential Operators

In the present paper we continue the project of systematic explicit construction of invariant differential operators. On the example of the non-compact group SO(p,9 - p) (for p = 5,6) we give the classification multiplets of indecomposable elementary representations induced from suitable choice of parabolic subgroup MAN so that the factor-group {G mathord{left/ {vphantom {G M}} right. kern-0em} M} is an Einstein manifold. This classification includes the data for the relevant invariant differential operators.

Jordan recoverability of some subcategories of modules over gentle algebras

Gentle algebras form a class of finite-dimensional algebras introduced by I. Assem and A. Skowroński in the 1980s. Modules over such an algebra can be described by string and band combinatorics in the associated gentle quiver from the work of M.C.R. Butler and C.M. Ringel. Any module can be naturally associated to a quiver representation. A nilpotent endomorphism of a quiver representation induces linear transformations over vector spaces at each vertex. Generically among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. We focus on subcategories additively generated by all the indecomposable representations of a gentle quiver, including a fixed vertex in their support. We show a characterization of the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.

Dimension vectors with the equal kernels property

Let r∈N, Kr be the generalized Kronecker quiver with r arrows γ1,…,γr:1→2 and δ∈Δ+(Kr) be a positive root of Kr. We say that δ has the equal kernels property if for all α∈kr∖{0} and every indecomposable representation M with dimension vector dim_M=δ the k-linear map Mα:=∑i=1rαiM(γi):M1→M2 is injective. We show that δ has the equal kernels property if and only if qKr(δ)+δ2−δ1≥1, where qKr:Z2→Z,(x,y)↦x2+y2−rxy denotes the Tits quadratic form of Kr.

Projective class ring of a restricted quantum group $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $

<abstract><p>In this paper, we compute the projective class ring of the new type restricted quantum group $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $. First, we describe the principal indecomposable projective $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $-modules and study their radicals, composition series, Cartan matrix of $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $ and so on. Then, we deconstruct the tensor products between two simple modules, two indecomposable projective modules and a simple module and an indecomposable projective module, into direct sum of some indecomposable representations. At last, we characterize the projective class ring by generators and relations explicitly.</p></abstract>

Dense Orbits of a Two Point Algebra

Finite dimensional algebras over a field can be classified into three classes by representation type. These are finite, tame, and wild. Wild algebras contain copies of the representations of all finite dimensional algebras so classifying the indecomposable representations is not feasible. We then look at representations whose orbit is dense in their associated irreducible component of their representation variety. With this geometric perspective, we can approximate our representations. We give an example of a two point dense orbit algebra of wild representation type and show that it has a dense orbit for dimension vectors of certain types.

Tiered Trees and Theta Operators

Abstract In [10], the authors introduced tiered trees to define combinatorial objects counting absolutely indecomposable representations of certain quivers and torus orbits on certain homogeneous varieties. In this paper, we use Theta operators, introduced in [6], to give a symmetric function formula that enumerates these trees. We then formulate a general conjecture that extends this result, a special case of which might give some insight about how to formulate a unified Delta conjecture [20].

Heisenberg Parabolic Subgroup of SO∗(10) and Invariant Differential Operators

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra so∗(10). We use the maximal Heisenberg parabolic subalgebra P=M⊕A⊕N with M=su(3,1)⊕su(2)≅so∗(6)⊕so(3). We give the main and the reduced multiplets of indecomposable elementary representations. This includes the explicit parametrization of the intertwining differential operators between the ERS. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebras so(p,q) (with p+q=10, p≥q≥2) with maximal Heisenberg parabolic subalgebra: P′=M′⊕A′⊕N′, M′=so(p−2,q−2)⊕sl(2,IR), M′C≅MC.

Categorification of Integer Sequences via Brauer Configuration Algebras and the Four Subspace Problem

The four subspace problem is a known matrix problem, which is equivalent to determining all the indecomposable representations of a poset consisting of four incomparable points. In this paper, we use solutions of this problem and invariants associated with indecomposable projective modules with some suitable Brauer configuration algebras to categorify the integer sequence encoded in the OEIS as A100705 and some related integer sequences.

On interval decomposability of 2D persistence modules

In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a wild problem. One direction is to consider the subclass of interval-decomposable persistence modules, which are direct sums of interval representations. We introduce the definition of pre-interval representations, a more natural algebraic definition, and study the relationships between pre-interval, interval, and thin indecomposable representations. We show that over the “equioriented” commutative 2D grid, these concepts are equivalent. Moreover, we provide a criterion for determining whether or not an nD persistence module is interval/pre-interval/thin-decomposable without having to explicitly compute decompositions. For 2D persistence modules, we provide an algorithm for determining interval-decomposability, together with a worst-case complexity analysis that uses the total number of intervals in an equioriented commutative 2D grid. We also propose several heuristics to speed up the computation.

PBW-Basis of Reduced Drinfeld Double Hall Algebra of Type An via Gröbner–Shirshov Basis

In the Ringel–Hall algebra of Dynkin type, the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Gröbner–Shirshov basis and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel–Hall algebra. We aim to generalize this result to the reduced Drinfeld double Hall algebra of type [Formula: see text]. First, we compute a minimal Gröbner–Shirshov basis for the reduced Drinfeld double Hall algebra of type [Formula: see text] by proving that all possible compositions between the commutator relations are trivial. Then, by taking the corresponding irreducible monomials, we construct a PBW-type basis for the reduced Drinfeld double Hall algebra of type [Formula: see text].

Continuous quivers of type A (I) foundations

We generalize quivers of type A to continuous quivers of type A and prove initial results about pointwise finite-dimensional representations. Among these results is the classification of those representations and a decomposition theorem, recovering results of Botnan and Crawley-Boevey (J Algebra Appl (2015). https://doi.org/10.1142/S0219498815500668, Decomposition of persistence modules, to appear in Proceedings of the American Mathematical Society, preprint: https://arxiv.org/pdf/1811.08946.pdf ). We also classify the indecomposable pointwise finite-dimensional projective representations. Finally, we prove that many of the properties of finite-dimensional representations of quivers of type $$A_n$$ also hold for finitely generated representations of continuous quivers of type A. This is the self-contained foundational part of a series of works to study a generalization of continuous clusters categories and their relationship to other cluster structures of type A.

On the Number of Indecomposable Modular Representations of a Cyclic p- Group over a Local Ring of Finite Length

On the Number of Indecomposable Modular Representations of a Cyclic p- Group over a Local Ring of Finite Length