Andrew Hubery
We consider the Kronecker quiver and determine the relations for the specialisation to q=0 of the generic composition algebra as well as those for Reineke's composition monoid. We also obtain a normal form for the varieties occurring in the composition monoid in terms of Schur roots.
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We compare various bases of the affine quantum group $\mathbfU^ (\hat\mathfrak{sl}_2)$ in the context of the Kronecker quiver, and relate them to the Drinfeld presentation.
We construct bar-invariant $\mathbb{Z}[q^{\pm 1/2}]-$bases of the quantum cluster algebra of the Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the cluster algebra of the Kronecker quiver in the sense of \cite{sherzel},\cite{calzel} and \cite{gls} respectively. As a byproduct, we prove the positivity of the elements in these bases.
We give a short introduction to Cohomological Hall algebras of quivers and describe the semistable Cohomological Hall algebra of central slope of the Kronecker quiver in terms of generators and relations.
An explicit description for quiver Grassmannians of the Kronecker quiver
本文针对箭图 Grassmannian 的胞腔分解公开问题，提出通过显式描述每个子表示的新方法，并以 Kronecker 箭图为例进行阐释。
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 study quiver Grassmannians associated with indecomposable representations of the Kronecker quiver. We find a cellular decomposition of them and we compute their Betti numbers. As an application, we give a geometric realization of the "canonical basis" of cluster algebras of Kronecker type (found by Sherman and Zelevinsky) and of type $A_2^{(1)}$.
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.
We deduce using the Ringel-Hall algebra approach explicit formulas for the cardinalities of some Grassmannians over a finite field associated to the Kronecker quiver. We realize in this way a quantification of the formulas obtained by Caldero and Zelevinsky for the Euler characteristics of these Grassmannians. We also present a recursive algorithm for computing the cardinality of every Kronecker quiver Grassmannian over a finite field.
We construct a Frobenius structure whose intersection form coincides with the generalized Cartan matrix of the $\ell$-Kronecker quiver $K_{\ell}$ and underlying complex manifold is isomorphic to the space of stability conditions for the bounded derived category of finitely generated modules over the path algebra $\mathbb{C} K_{\ell}$.
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 …
A set valued representation of the Kronecker quiver is nothing but a quiver. We apply the forgetful functor from vector spaces to sets and compare linear with set valued representations of the Kronecker quiver.
Let Q be a quiver. M. Reineke and A. Hubery investigated the connection between the composition monoid, as introduced by M. Reineke, and the generic composition algebra, as introduced by C. M. Ringel, specialised at q=0. In this thesis we continue their work. We show that if Q is a Dynkin quiver or an oriented cycle, then the composition algebra at q=0 is isomorphic to the monoid algebra of the composition monoid. Moreover, if Q is an acyclic, extended Dynkin quiver, we show that there exists an epimorphism from the composition algebra at q=0 to the monoid algebra of the composition monoid, an…
We study Euler characteristics of moduli spaces of stable representations of m-Kronecker quivers for m>>0.
We apply the notion of hyperfinite families of modules to the wild path algebras of generalised Kronecker quivers $kΘ(d)$. While the preprojective and postinjective component are hyperfinite, we show the existence of a family of non-hyperfinite modules in the regular component for some $d$. Making use of dimension expanders to achieve this, our construction is more explicit than previous results. From this it follows that no finitely controlled wild algebra is of amenable representation type.
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 $Q$ 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…
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…
Motivated by string-theoretic arguments Manschot, Pioline and Sen discovered a new remarkable formula for the Poincare polynomial of a smooth compact moduli space of stable quiver representations which effectively reduces to the abelian case (i.e. thin dimension vectors). We first prove a motivic generalization of this formula, valid for arbitrary quivers, dimension vectors and stabilities. In the case of complete bipartite quivers we use the refined GW/Kronecker correspondence between Euler characteristics of quiver moduli and Gromov-Witten invariants to identify the MPS formula for Euler cha…
Geometric realization of affine bases: the Kronecker quiver case
本文从几何角度研究Kronecker箭图量子包络代数负部分中PBW基与标准基之间的转移矩阵，通过旗层复形构造PBW基元素的几何实现，并证明转移系数由交上同调复形限制到小子层的局部系统重数决定。
We study a monoid associated to complex semisimple Lie algebras, called the quantic monoid. Its monoid ring is shown to be isomorphic to a degenerate quantized enveloping algebra. Moreover, we provide normal forms and a straightening algorithm for this monoid. All these results are proved by a realization in terms of representations of quivers, namely as the monoid of generic extensions of a quiver with automorphism.
Let $Q$ be a wild $n$-Kronecker quiver, i.e., a quiver with two vertices, labeled by 1 and 2, and $n\geq 3$ arrows from 2 to 1. The indecomposable regular modules with preprojective Gabriel-Roiter submodules, in particular, those $τ^{-i}X$ with $\udim X=(1,c)$ for $i\geq 0$ and some $1\leq c\leq n-1$ will be studied. It will be shown that for each $i\geq 0$ the irreducible monomorphisms starting with $τ^{-i}X$ give rise to a sequence of Gabriel-Roiter inclusions, and moreover, the Gabriel-Roiter measures of those produce a sequence of direct successors. In particular, there are infinitely many…
We investigate the generalized Kronecker algebra $\mathcal{K}_r = kΓ_r$ with $r \geq 3$ arrows. Given a regular component $\mathcal{C}$ of the Auslander-Reiten quiver of $\mathcal{K}_r$, we show that the quasi-rank $rk(\mathcal{C}) \in \mathbb{Z}_{\leq 1}$ can be described almost exactly as the distance $\mathcal{W}(\mathcal{C}) \in \mathbb{N}_0$ between two non-intersecting cones in $\mathcal{C}$, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality \[ -\mathcal{W}(\mathcal{C}) \leq rk(\mathcal{C}) \le…
Gromov-Witten invariants of weighted projective planes and Euler characteristics of moduli spaces of representations of bipartite quivers are related via the tropical vertex, a group of formal automorphisms of a torus. On the Gromov-Witten side, this uses the work of Gross, Pandharipande and Siebert. The quiver moduli side features quiver wall-crossing formulas, functional equations for Euler characteristics, and localization techniques. We derive several explicit formulas for Gromov-Witten invariants.
We prove that the semi-invariant ring of the standard representation space of the $l$-flagged $m$-arrow Kronecker quiver is an upper cluster algebra for any $l,m\in \mathbb{N}$. The quiver and cluster are explicitly given. We prove that the quiver with its rigid potential is a polyhedral cluster model. As a consequence, to compute each Kronecker coefficient $g_{μ,ν}^λ$ with $λ$ at most $m$ parts, we only need to count lattice points in at most $m!$ fibre (rational) polytopes inside the ${\rm g}$-vector cone, which is explicitly given.
If $R$ is a finite commutative ring, then the affine monoid of $R$ is the monoid of all affine mappings $x\mapsto ax+b$ on $R$. Alternatively, it is the semidirect product of the multiplicative monoid of $R$ with the additive group of $R$. In this paper we compute the Gabriel quiver of the complex algebra of the affine monoid of any finite commutative ring.
We give a construction of the projective indecomposable modules and a description of the quiver for a large class of monoid algebras including the algebra of any finite monoid whose principal right ideals have at most one idempotent generator. Our results include essentially all families of finite monoids for which this has been done previously, for example, left regular bands, $\mathscr J$-trivial and $\mathscr R$-trivial monoids and left regular bands of groups.
Dimension vectors of elementary modules of generalized Kronecker quivers
本文研究了广义 $n$-Kronecker 箭图初等模的维数向量，构造了特定的初等模 $X(x,y)$，并证明了初等模维数分量的上界条件及特定维数下成为初等模的充要条件。
We study Laurent expansions of cluster variables in a cluster algebra of rank 2 associated to a generalized Kronecker quiver. In the case of the ordinary Kronecker quiver, we obtain explicit expressions for Laurent expansions of the elements of the canonical basis for the corresponding cluster algebra.
Monoidal seeds of the categories $\mathcal{C}_{wv}$ over quiver Hecke algebras
针对对称箭图Hecke代数，利用反射函子与新算子构造了$\boldsymbol{\frak{C}_{wv}}$的量子幺子种子，证明其与特定行列式模的素因子集合一致，且$\boldsymbol{\frak{C}_{wv}}$的Grothendieck环介于簇代数与上簇代数之间。
Combining the MPS degeneration formula for the Poincaré polynomial of moduli spaces of stable quiver representations and localization theory, it turns that the determination of the Euler characteristic of these moduli spaces reduces to a combinatorial problem of counting certain trees. We use this fact in order to obtain an upper bound for the Euler characteristic in the case of the Kronecker quiver. We also derive a formula for the Euler characteristic of some of the moduli spaces appearing in the MPS degeneration formula.
Solution of a Problem in Monoidal Categorification by Additive Categorification
本文利用Palu广义变异规则和整体维数至多2的代数相关簇范畴，为Kashiwara-Kim-Oh-Park构造的单项范畴化种子提供了显式箭图，从而解决了量子群表示论与箭图表示论之间的一个簇组合学问题。
We compute the quiver of any monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as $\mathbf{DO}$) to representation theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this…
For the Kronecker quiver, Zwara has found an example of a representation whose orbit closure is neither unibranch nor Cohen-Macaulay. In this note, we explain how to extend this example to all infinite type quivers without oriented cycles.
We introduce certain quiver analogue of the determinantal variety. We study the Kempf-Lascoux-Weyman's complex associated to a line bundle on the variety. In the case of generalized Kronecker quivers, we give a sufficient condition on when the complex resolves a maximal Cohen-Macaulay module supported on the quiver determinantal variety. This allows us to find the set-theoretical defining equations of these varieties. When the variety has codimension one, the only irreducible polynomial function is a relative tensor invariant. As a by-product, we find some vanishing condition for the Kronecker…
In this note, we introduce monoidal subcategories of the tensor category of finite-dimensional representations of a simply-laced quantum affine algebra, parametrized by arbitrary Dynkin quivers. For linearly oriented quivers of types A and D, we show that these categories provide monoidal categorifications of cluster algebras of the same type. The proof is purely representation-theoretical, in the spirit of [arXiv:0903.1452].
The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part U_q(n) of the universal enveloping algebra of a Kac-Moody Lie algebra of type A_1^{(1)}. The examined subalgebra is associated with a terminal module M over the path algebra of the Kronecker quiver via an Weyl group element w of length four. Geiss-Leclerc-Schroeer attached to M a category C_M of nilpotent modules over the preprojective algebra of the Kronecker quiver together with an acyclic cluster algebra A(C_M). The dual semicanonical basis contains all cluster monomials. By construction, the clu…
The (ordinary) quiver of an algebra $A$ is a graph that contains information about the algebra's representations. We give a description of the quiver of $\mathbb{C}PT_{n}$, the algebra of the monoid of all partial functions on $n$ elements. Our description uses an isomorphism between $\mathbb{C}PT_{n}$ and the algebra of the epimorphism category, $E_{n}$, whose objects are the subsets of $\{1,\ldots, n\}$ and morphism are all total epimorphisms. This is an extension of a well known isomorphism of the algebra of $IS_{n}$ (the monoid of all partial injective maps on $n$ elements) and the algebra…
Graded Kronecker Modules of Constant Rank
本文研究广义Kronecker箭图万有覆盖上的常秩表示，证明常秩表示子范畴等于单射表示子范畴与满射表示子范畴之并。
Let kK be the path algebra of the Kronecker quiver and consider the category of finite dimensional right modules over kK (called Kronecker modules). We prove that extensions of Kronecker modules are field independent up to Segre classes, so they can be described purely combinatorially. We use in the proof explicit descriptions of particular extensions and a variant of the well known Green formula for Ringel-Hall numbers, valid over arbitrary fields. We end the paper with some results on extensions of preinjective Kronecker modules, involving the dominance ordering from partition combinatorics …
We consider the Kronecker algebra $A=\mathcal{O}[X, Y]/(X^2, Y^2)$, where $\mathcal{O}$ is a complete discrete valuation ring. Since $A \otimesκ$ is a special biserial algebra, where $κ$ is the residue field of $\mathcal{O}$, one can compute a complete list of indecomposable $A\otimesκ$-modules. For each indecomposable $A\otimesκ$-module, we obtain a special kind of $A$-lattices called "Heller lattices". In this paper, we determine the non-periodic component of a variant of the stable Auslander--Reiten quiver for the category of $A$-lattices that contains "Heller lattices". Moreover, we give t…
Representations of Kronecker quivers and Steiner bundles on Grassmannians
本文利用Auslander-Reiten理论和反射函子研究Grassmann流形上Steiner丛与广义Kronecker箭图相对投射表示满子范畴之间的对应关系，并开创了对野表示范畴中与初等测试模右Hom-正交的满子范畴的研究。
Quivers of stylic algebras
本文为 stylic 幺半群代数构造了本原正交幂等元完备系统，并给出了其显式箭图表示。
Elementary modules of the generalized Kronecker quivers K(n)
本文研究广义 Kronecker 箭图 K(n) 的基本正则模，在基本域 F 中当 x<n 时建立了基本模与固定秩空间的一一对应关系。
We prove that the localization of the monoidal category $\mathcal{C}_w$ is rigid, and the category $\mathcal{C}_{w,v}$ admits a localization via a real commuting family of central objects. Note that the localization of $\mathcal{C}_{w,v}$ categorifies the open Richardson variety.
We study a compatibility relationship between Qin's dominance order on a cluster algebra $\mathcal{A}$ and partial orderings arising from classifications of simple objects in a monoidal categorification $\mathcal{C}$ of $\mathcal{A}$. Our motivating example is Hernandez-Leclerc's monoidal categorification using representations of quantum affine algebras. In the framework of Kang-Kashiwara-Kim-Oh's monoidal categorification via representations of quiver Hecke algebras, we focus on the case of the category $R-gmod$ for a symmetric finite type $A$ quiver Hecke algebra using Kleshchev-Ram's classi…
Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra. In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of…
We provide the localization procedure for monoidal categories by a real commuting family of braiders. For an element $w$ of the Weyl group, $\mathscr{C}_w$ is a subcategory of modules over quiver Hecke algebra which categorifies the quantum unipotent coordinate algebra $A_q[\mathfrak{n}(w)]$. We construct the localization $\widetilde{\mathscr{C}_w}$ of $\mathscr{C}_w$ by adding the inverses of simple modules which correspond to the frozen variables in the quantum cluster algebra $A_q[\mathfrak{n}(w)]$. The localization $\widetilde{\mathscr{C}_w}$ is left rigid and we expect that it is rigid.
Motives of central slope Kronecker moduli
利用反射函子诱导的箭图模空间对偶性，将中心斜率Kronecker模空间的 motive 生成级数描述为代数方程和 q-差分方程的解。
We study finitary 2-categories associated to dual projection functors for finite dimensional associative algebras. In the case of path algebras of admissible tree quivers (which includes all Dynkin quivers of type A) we show that the monoid generated by dual projection functors is the Hecke-Kiselman monoid of the underlying quiver and also obtain a presentation for the monoid of indecomposable subbimodules of the identity bimodule.
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…
Let $A$ be a finite-dimensional algebra over an algebraically closed field $\Bbbk$. For any finite-dimensional $A$-module $M$ we give a general formula that computes the indecomposable decomposition of $M$ without decomposing it, for which we use the knowledge of AR-quivers that are already computed in many cases. The proof of the formula here is much simpler than that in a prior literature by Dowbor and Mróz. As an example we apply this formula to the Kronecker algebra $A$ and give an explicit formula to compute the indecomposable decomposition of $M$, which enables us to make a computer prog…
We show that the generic Hall algebra of nilpotent representations of an oriented cycle specialised at $q=0$ is isomorphic to the generic extension monoid in the sense of Reineke. This continues the work of Reineke.
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…
We study the monoid of so called projection functors $\p{S}$ attached to simple modules $S$ of a finite dimensional algebra, which appear naturally in the study of torsion pairs. We determine defining relations in special cases of path algebras. For the linearly oriented Dynkin quiver of Type $A$, we get an isomorphism to the monoid of non-decreasing parking functions. Moreover we give an explicit isomorphism between the monoid algebra of non-decreasing parking functions and a certain incidence algebra independent of the field.
We relate the $m$-truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when $m=2$. Each {\sf g}-vector cone ${\sf G}_{\Diamond_l}$ of these cluster algebras controls the $2$-truncated Kronecker products for all symmetric functions of degree no greater than $l$. As a consequence, each relevant Kronecker coefficient is the difference of the number of the lattice points inside two rational polytopes. We also give explicit description of all ${\sf G}_{\Diamond_l}$'s. …
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。
Shift orbits for elementary representations of Kronecker quivers
本文研究野 $r$-Kronecker 箭图初等表示的维向量，利用 $G_r$ 移位作用的基本域给出判别不等式，并证明 $r\geq4$ 不存在类似 $K_3$ 的简洁组合描述。
The representation theory of the wreath product of a finite group with the monoid of all partial functions on a finite set as an EI-category algebra
通过分析相关Ehresmann EI-范畴，描述了有限群G与部分函数幺半群PT_n的圈积的复幺半群代数的通常箭图，并证明了其整体维数为n-1。
Quivers with quantum Yang-Baxter equation and Hecke condition: Deformation of face algebras
本文引入带量子Yang-Baxter和Hecke结构的箭图，研究其路代数中由Hecke R-矩阵确定的RTT关系，证明量子矩阵代数与玫瑰箭图上的面代数同构。
We study the growth of the ground state degeneracy in the Kronecker model of quiver quantum mechanics. This is the simplest quiver with two gauge groups and bifundamental matter fields, and appears universally in the context of BPS state counting in four-dimensional N=2 systems. For large ranks, the ground state degeneracy is exponential with slope a modular function that we are able to compute at integral values of its argument. We also observe that the exponential of the slope is an algebraic number and determine its associated algebraic equation explicitly in several examples. The speed of …
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