Giovanni Cerulli Irelli
This paper contains the material discussed in the series of three lectures that I gave during the workshop of the ICRA 2018 in Prague. I will introduce the reader to some of the techniques used in the study of the geometry of quiver Grassmannians. The notes are quite elementary and thought for phd students or young researchers. I assume that the reader is familiar with the representation theory of quivers.
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Caldero and Zelevinsky studied the geometry of quiver Grassmannians for the Kronecker quiver and computed their Euler characteristics by examining natural stratification of quiver Grassmannians. We consider generalized Kronecker quivers and compute virtual Poincare polynomials of certain varieties which are the images under projections from strata of quiver Grassmannians to ordinary Grassmannians. In contrast to the Kronecker quiver case, these polynomials do not necessarily have positive coefficients. The key ingredient is the explicit formula for noncommutative cluster variables given by Ral…
The paper includes a new proof of the fact that quiver Grassmannians associated with rigid representations of Dynkin quivers do not have cohomology in odd degrees. Moreover, it is shown that they do not have torsion in homology. A new proof of the Caldero--Chapoton formula is provided. As a consequence a new proof of the positivity of cluster monomials in the acyclic clusters associated with Dynkin quivers is obtained. The methods used here are based on joint works with Markus Reineke and Evgeny Feigin.
Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are homeomorphic to the lagrangian quiver varieties of Nakajima which have been well studied in the context of geometric representation theory. We then refine this result by finding quiver grassm…
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…
A desingularization of arbitrary quiver Grassmannians for representations of Dynkin quivers is constructed in terms of quiver Grassmannians for an algebra derived equivalent to the Auslander algebra of the quiver.
In this text, we characterize the representation type of an acyclic quiver by the properties of its associated quiver Grassmannians. This characterization utilizes and extends known results about singular quiver Grassmannians and cell decompositions into affine spaces. While all quiver Grassmannians for indecomposable representations of quivers of finite representation types $A$ and $D$ are smooth and admit cell decompositions, it turns out that all quiver Grassmannians for indecomposable representations of quivers of tame types $A$ and $D$ admit cell decompositions, but some of these quiver G…
Inspired by recent work of Cerulli-Feigin-Reineke on desingularizations of quiver Grassmannians of representations of Dynkin quivers, we obtain desingularizations in considerably more general situations and in particular for Grassmannians of modules over iterated tilted algebras of Dynkin type. Our desingularization map is constructed from Nakajima's desingularization map for graded quiver varieties.
Quiver Grassmannians are projective varieties parametrizing subrepresentations of given dimension in a quiver representation. We define a class of quiver Grassmannians generalizing those which realize degenerate flag varieties. We show that each irreducible component of the quiver Grassmannians in question is isomorphic to a Schubert variety. We give an explicit description of the set of irreducible components, identify all the Schubert varieties arising, and compute the Poincaré polynomials of these quiver Grassmannians.
The product of simple modules over KLR algebras and quiver Grassmannians
本文通过箭图格拉斯曼流形研究 KLR 代数上单模乘积，建立归纳函子与不可约分支的联系，给出乘积为单模的必要条件，并证明 Lapid-Minguez 猜想的部分情形。
We show that any quiver Grassmannian associated with a rigid representation of a quiver is a rational variety using torus localization techniques.
This is an extended abstract of my talk at the Oberwolfach Workshop "Interactions between Algebraic Geometry and Noncommutative Algebra" (May 10 - 14, 2010). We present some properties of quiver Grassmannians and examples of explicit computations of their Euler characteristics.
Desingularizations of Quiver Grassmannians for the Equioriented Cycle Quiver
该研究针对等向循环箭图的幂零表示，构造了箭图格拉斯曼流形的环面等变去奇异化方法，并将其用于计算相应的环面等变上同调。
In this text, we exhibit the quiver Plücker relations for a quiver Grassmannian and show that they describe the quiver Grassmannian as a closed subscheme of a product of usual Grassmannians.
For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_q^-(\mathfrak{g})$, where $U_q^-(\mathfrak{g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig's category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result…
In this paper, we introduce Schubert decompositions for quiver Grassmannians and investigate example classes of quiver Grassmannians with a Schubert decomposition into affine spaces. The main theorem puts the cells of a Schubert decomposition into relation to the cells of a certain simpler quiver Grassmannian. This allows us to extend known examples of Schubert decompositions into affine spaces to a larger class of quiver Grassmannians. This includes exceptional representations of the Kronecker quiver as well as representations of forests with block matrices of the form $\begin{pmatrix} 0&1\\ …
PrIncipal quiver Grassmannians: conjectures
本文通过计算机实验提出关于引物格拉斯曼流形代数几何性质的猜想，探讨子表示的结构与维度关系。
Degenerate Affine Flag Varieties and Quiver Grassmannians
本文通过循环箭图的箭图格拉斯曼流形研究退化仿射旗簇的有限维逼近，证明了其胞腔分解及不可约分量的正规性与有理奇点性质。
Postnikov constructed a cellular decomposition of the totally nonnegative Grassmannians. The poset of cells can be described (in particular) via Grassmann necklaces. We study certain quiver Grassmannians for the cyclic quiver admitting a cellular decomposition, whose cells are naturally labeled by Grassmann necklaces. We show that the posets of cells coincide with the reversed cell posets of the cellular decomposition of the totally nonnegative Grassmannians. We investigate algebro-geometric and combinatorial properties of these quiver Grassmannians. In particular, we describe the irreducible …
We study the connection between the affine degenerate Grassmannians in type $A$, quiver Grassmannians for one vertex loop quivers and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type $GL_n$ and identify it with semi-infinite orbit closure of type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex loop quiver provide finite-dimensional approximations of the degenerate affine Grassmannian. Finally, we give an explicit description of the degenerate affine Grassmannian of type $A_1^{(1)}$, propose a conjectural descrip…
We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew $(GL(m), GL(n))$ duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine…
Quasi-projective varieties are Grassmannians for fully exact subcategories of quiver representations
本文通过限制Reineke同构到完全正合子范畴，证明拟射影簇可表示为箭图格拉斯曼流形。
We construct Nakajima's quiver varieties of type A in terms of conjugacy classes of matrices and (non-Slodowy's) transverse slices naturally arising from affine Grassmannians. In full generality quiver varieties are embedded into Beilinson-Drinfeld Grassmannians of type A. Our construction provides a compactification of Nakajima's quiver varieties and a decomposition of an affine Grassmannian into a disjoint union of quiver varieties. As an application we provide a geometric version of skew and symmetric $(GL(m), GL(n))$ duality.
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 Grassmanians 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, …
Quiver Grassmannians for the Bott-Samelson resolution of type A Schubert varieties
本文将A型Schubert簇的Bott-Samelson解消实现为箭图Grassmannian，通过引入置换的几何相容分解概念和构造带关系的特殊箭图，显式建立了这两者之间的同构关系。
We show that the cohomology ring of a quiver Grassmannian asssociated with a rigid quiver representation has property (S): there is no odd cohomology and the cycle map is an isomorphism; moreover, its Chow ring admits explicit generators defined over any field. From this we deduce the polynomial point count property. By restricting the quiver to finite or affine type, we are able to show a much stronger assertion: namely, that a quiver Grassmannian associated to an indecomposable (not necessarily rigid) representation admits a cellular decomposition. As a corollary, we establish a cellular dec…
Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proven that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits, and a cell…
Tropical Quiver Grassmannians
本文通过引入赋值拟阵的箭图与箭图Dressian，研究热带线性空间包含关系的参数化，并证明从环境维数2起箭图Dressian存在不可实现点。
Flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology
本文证明了Dynkin箭图（A、D、E型）的箭图Grassmannian旗版本无奇上同调，A和D型具仿射铺砌，并推广了刚性表示的对角分解结果。
We present a review on the recently discovered link between the Lie theory,the theory of quiver Grassmannians, and various degenerations of flag varieties. Our starting point is the induced Poincaré--Birkhoff--Witt filtration on the highest weight representations and the corresponding PBW degenerate flag varieties.
Skeletal Torus Actions and GKM Structures on Quiver Grassmannians of String Representations
本文研究等定向A型与幂零等定向Ã型箭图表示的箭图格拉斯曼流形，证明其为GKM簇，并对弦表示情形下具有GKM簇结构的所有相应箭图格拉斯曼流形给出完整分类。
Quiver matroids -- Matroid morphisms, quiver Grassmannians, their Euler characteristics and $\mathbb{F}_1$-points
本文基于有向图初等因子与映射，研究其欧拉特征及F₁点，通过推广为束并构建模空间，实现复有向图Grassmannian的F₁类比。
We prove a decomposition theorem for irreducible components of Grassmannians of submodules, as well as for other schemes arising from representation theory, thus generalising the result of Crawley-Boevey and Schroer for module varieties. The method is based on jet space calculations, using that the formation of direct sums induces a separable morphism of schemes.
This is an expanded version of lectures given at a Summer School "Geometric methods in Representation Theory" (Grenoble, 2008).
We propose a conjectural construction of various slices for double affine Grassmannians as Coulomb branches of 3-dimensional N=4 supersymmetric affine quiver gauge theories. It generalizes the known construction for the usual affine Grassmannian, and makes sense for arbitrary symmetric Kac-Moody algebras.
Extending the main result of Part 1, in the first part of this paper we show that every quiver Grassmannian of a representation of a quiver of extended Dynkin type $D$ has a decomposition into affine spaces. In the case of real root representations of small defect, the non-empty cells are in one-to-one correspondence to certain, so called non-contradictory, subsets of the vertex set of a fixed tree-shaped coefficient quiver. In the second part, we use this characterization to determine the generating functions of the Euler characteristics of the quiver Grassmannians (resp. $F$-polynomials). Al…
Laumon parahoric local models via quiver Grassmannians
本文利用箭图表示理论研究类型 A 中任意仿抛物子群的 Laumon 局部模型特殊纤维，刻画其不可约分支、自然分层和胞腔分解，并证明奇点消解与仿抛物子群间自然投影相容。
We unify aspects of the equivariant geometry of type $D$ quiver representation varieties, double Grassmannians, and symmetric varieties $GL(a+b)/GL(a)\times GL(b)$; in particular we translate results about singularities of orbit closures, combinatorics of orbit closure containment, and torus equivariant $K$-theory between these three families. These results are all obtained from our generalization of a construction of Zelevinsky for type $A$ quivers to the type $D$ setting. More precisely, we give explicit embeddings with nice properties of homogeneous fiber bundles over type $D$ quiver repres…
Extremality of principal quiver Grassmannians
研究箭图格拉斯曼流形族，通过明确描述其平坦不可约轨迹，证明该族成员不可约且具预期维度的充要条件是周围箭图表示退化为特定直和。
Specialization map for quiver Grassmannians
本文定义了Dynkin型箭图Grassmann流形上同调代数间的特化映射，并证明其在A型中满射，推广了Lanini和Strickland的结果。
Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler characteristics. Our main result is the positivity of Euler characteristics when M is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras.
Symplectic Grassmannians and Cyclic Quivers
本文将箭图Grassmann流形方法推广到辛Grassmann簇退化情形，构造胞腔分解、计算Euler示性数与Poincaré多项式，并证明不可约分支数等于经典辛Grassmann流形的Euler示性数。
On quiver Grassmannians and orbit closures for gen-finite modules
本文通过典范倾斜模与再收集构造gen-有限模的轨道闭包及箭图格拉斯曼流形的去奇异化，推广了2017年相关结果，并给出对偶版本以处理cogen-有限模。
Loop Grassmannian of quivers and Compactified Coulomb branch of quiver gauge theory with no framing
本文证明当对称整数矩阵取为箭图Q的关联矩阵时，Mirković的环路格拉斯曼构造与无框架箭图规范理论的紧化库仑分支作为Zastava的两种推广相互同构。
Let $Q$ be a quiver, $M$ a representation of $Q$ with an ordered basis $\cB$ and $\ue$ a dimension vector for $Q$. In this note we extend the methods of \cite{L12} to establish Schubert decompositions of quiver Grassmannians $\Gr_\ue(M)$ into affine spaces to the ramified case, i.e.\ the canonical morphism $F:T\to Q$ from the coefficient quiver $T$ of $M$ w.r.t.\ $\cB$ is not necessarily unramified. In particular, we determine the Euler characteristic of $\Gr_\ue(M)$ as the number of \emph{extremal successor closed subsets of $T_0$}, which extends the results of Cerulli Irelli (\cite{Cerulli11…
We construct for each choice of a quiver $Q$, a cohomology theory $A$ and a poset $P$ a "loop Grassmannian" $\mathcal{G}^P(Q,A)$. This generalizes loop Grassmannians of semisimple groups and the loop Grassmannians of based quadratic forms. The addition of a dilation torus $\mathcal{D}\subset\mathbb{G}_m^2$ gives a quantization $\mathcal{G}^P_\mathcal{D}(Q,A)$. The construction is motivated by the program of introducing an inner cohomology theory in algebraic geometry adequate for the Geometric Langlands program ['Some extensions of the notion of loop Grassmannians', Rad Hrvat. Akad. Znan. Umje…
It is a well established fact, that any projective algebraic variety is a moduli space of representations over some finite dimensional algebra. This algebra can be chosen in several ways. The counterpart in algebraic geometry is tautological: every variety is its own Hilber scheme of sheaves of length one. This holds even scheme theoretic. We use Beilinson's equivalence to get similar results for finite dimensional algebras, including moduli spaces and quiver grassmannians. Moreover, we show that several already known results can be traced back to the Hilbert scheme construction and Beilinson'…
These lecture notes consist of an introduction to moduli spaces in algebraic geometry, with a strong emphasis placed on examples related to the theory of quiver representations. The goal is to provide the background necessary to understand the construction of the moduli spaces of stable and semistable quiver representations due to King, Nakajima's quiver varieties, as well as some 'quiver versions' of existing moduli spaces of algebra-geometric objects.
In this article, we use the relationship between cluster scattering diagrams and stability scattering diagrams to relate quiver representations with these diagrams. With a notion of positive crossing of a path $γ$, we show that if $γ$ has positive crossing in the scattering diagram, then it goes in the opposite direction of the Auslander-Reiten quiver of $Q$. We then give the Hall algebra theta functions which recover the cluster character formula by the Euler characteristic map. At last, we define the Hall algebra broken lines and then are able to give the stratification of the quiver Grassma…
Euler Characteristics of Generic Quiver Grassmannians: Semi-Invariants and Localization
研究有限无圈箭图\(Q\)的一般箭图格拉斯曼流形\(\operatorname{Gr}_\beta(V)\)，通过结合多种公式将其拓扑欧拉特征表示为协变重数的有限带符号和，给出任意\(Q\)的有限环面局部化公式，并在广义克罗内克箭图情形下进行说明。
Quiver Grassmannians associated to nilpotent cyclic representations defined by single matrix
本文研究由单个矩阵定义的幂零循环表示所关联的箭图Grassmannian的闭Białynicki-Birula胞腔几何，证明在子表示维数为全1时这些胞腔光滑，并借助Knutson-Tao基描述了其上同调环的乘法结构。
In this work we study a realization of moduli spaces of framed quiver representations as Grassmannians of submodules devised by Marcus Reineke. Obtained is a generalization of this construction for finite dimensional associative algebras and for quivers with oriented cycles. As an application we get an explicit realization of fibers for the moduli space bundle over the categorical quotient for the quiver $A_{n-1}^{(1)}$.
Let $Q$ be a quiver of extended Dynkin type $D$. In this first of two papers, we show that the quiver Grassmannian $Gr_e(M)$ has a decomposition into affine spaces for every dimension vector $e$ and every indecomposable representation $M$ of defect $-1$ and defect $0$, with exception of the non-Schurian representations in homogeneous tubes. We characterize the affine spaces in terms of the combinatorics of a fixed coefficient quiver for $M$. The method of proof is to exhibit explicit equations for the Schubert cells of $Gr_e(M)$ and to solve this system of equations successively in linear term…
From brick manifolds to Grassmannians of bimodules
本文研究箭图路径代数上子双模的格拉斯曼流形，推广了砖流形，给出显式构造、光滑性证明、胞腔分解及带框模空间实现，并导出动机的递归公式。
Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity
研究从射影直线到拉格朗日格拉斯曼流形的基于代数映射空间，通过箭图描述实现代数几何精细化，得到同构，其拓扑实现恢复经典同伦等价。
We show that a semi-stable moduli space of representations of an acyclic quiver can be identified with two GIT quotients by reductive groups. One of a quiver Grassmannian of a projective representation, the other of a quiver Grassmannian of an injective representation. This recovers as special cases the classical Gelfand-MacPherson correspondence and its generalization by Hu and Kim to bipartite quivers, as well as the Zelevinsky map for a quiver of Dynkin type A with the linear orientation.
We study degenerations of Grassmannians constructed using convex lattice configurations in Bruhat-Tits buildings. Using techniques from quiver representations, we analyze their special fibers, which are explicitly described as quiver Grassmannians. For a class of lattice configurations, called the locally linearly independent configurations, we show that our construction coincide with Mustafin degenerations, thus generalizing a result of Faltings. In such cases, our analysis of special fibers also generalizes results of Cartwright et al. As an application, we prove a smoothing criterion for li…
We introduce a construction that associates, to each finite dimensional k-vector space V, a family of projective k-varieties that comes equipped with the structure of a operad in the category of k-schemes. When dim V = 1, this operad contains, as a suboperad, the family of moduli spaces of stably marked rational curves. For general V, our operad contains the family of Chen-Gibney-Krashen moduli spaces of stably marked trees of projective spaces as a suboperad. We realize our operad as part of a larger theory that describes how to construct operads from suitable functors. Given a category C tha…
This is the written version of my five lectures at the Banach Center mini-school on "Schubert Varieties", in Warsaw, May 18-22, 2003.
This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type $ADE$. In the unframed case they are isomorphic to the moduli space of based rational maps from ${\mathbb C}P^1$ to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian.
We give a complete quiver description of the category of perverse sheaves on Hermitian symmetric spaces in types A and D, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety.
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