Cell decompositions of quiver flag varieties for nilpotent representations of the oriented cycle
Julia Sauter
备注：
10 pages. Minor revision because the modules in my application are not examples of the modules studied by Kato in his work on geometric extension algebras
Generalizing Schubert cells in type A and a cell decomposition if Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of an oriented cycle admit an affine cell decomposition parametrized by multi-tableaux. We show that they carry a torus operation and describe the T-equivariant cohomology using Goresky-Kottwitz-MacPherson-theory. As an application of the cell decomposition we obtain a vector space basis of certain modules (for quiver Hecke algebras of nilpotent representations of this quiver), similar modules have been studied by Kato as analogues of standard modules.
We define a set of "enhanced" nilpotent quiver representations that generalizes the enhanced nilpotent cone. This set admits an action by an associated algebraic group $K$ with finitely many orbits. We define a combinatorial set that parametrizes the set of orbits under this action and we derive a purely combinatorial formula for the dimension of an orbit.
We prove that the local intersection cohomology of nilpotent orbit closures of cyclic quivers is trivial when the two orbits involved correspond to partitions with at most two rows. This gives a geometric proof of a result of Graham and Lehrer, which states that standard modules of the affine Hecke algebra of $GL_d$ corresponding to nilpotents with at most two Jordan blocks are multiplicity-free.
This paper presents analogous results of Hua [7][8] on numbers of representations of quivers over finite fields which respect nilpotent relations under certain assumptions. A closed formula which counts isomorphism classes of absolutely indecomposable representations with given dimension vectors is given and a $q$-deformation of Weyl-Kac denominator identity is established. In principle, if the numbers of representations are known, then the numbers of isomorphism classes of absolutely indecomposable representations are known.
We gave a classification of P and Q with a finite number of K-orbits of a double flag variety G/P*K/Q for a symmetric pair (G, K) when G=GL_{m+n} and K=GL_{m}*GL_{n}, and a description of K-orbits when the number of K-orbits of G/P*K/Q is finite. We solved the problem by providing a correspondence between the K-orbits and the quiver representations.
The intersection cohomologies of closures of nilpotent orbits of linear (respectively, cyclic) quivers are known to be described by Kazhdan-Lusztig polynomials for the symmetric group (respectively, the affine symmetric group). We explain how to simplify this description using a combinatorial cancellation procedure, and derive some consequences for representation theory.
We give a closed expression for the number of points over finite fields (or the motive) of the Lusztig nilpotent variety associated to any quiver, in terms of Kac's A-polynomials. When the quiver has 1-loops or oriented cycles, there are several possible variants of the Lusztig nilpotent variety, and we provide formulas for the point count of each. This involves nilpotent versions of the Kac A-polynomial, which we introduce and for which we give a closed formula similar to Hua's formula for the usual Kac A-polynomial. Finally we compute the number of points over a finite field of the various s…
The $\mathrm{GL}(V)$-orbits in the enhanced nilpotent cone $V\times\mathcal{N}(V)$ are (essentially) in bijection with the orbits of a certain parabolic $P\subseteq\mathrm{GL}(V)$ (the mirabolic subgroup) in the nilpotent cone $\mathcal{N}(V)$. We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameterization generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different to the previous ones that have appeared in the literature. Explicit translations between the different pa…
This paper introduces a class of smooth projective varieties that generalise and share many properties with partial flag varieties of type A. The quiver flag variety M_\vartheta(Q,r) of a finite acyclic quiver Q (with a unique source) and a dimension vector r is a fine moduli space of stable representations of Q. Quiver flag varieties are Mori Dream Spaces, they are obtained via a tower of Grassmann bundles, and their bounded derived category of coherent sheaves is generated by a tilting bundle. We define the multigraded linear series of a weakly exceptional sequence of locally free sheaves E …
We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type $\tt A$ flag varieties, their linear degenerations, finite dimensional approximations of the $GL_n$-affine flag variety and affine Grassmannian. We show that these quiver Grassmannians, equipped with our torus action, are GKM-varieties and that their moment graph admits a combinatorial description in terms of coefficients quiver of the underlying quiver representations. By adapting to our setting results by Gonzales, we are able …
The flag variety of a complex reductive linear algebraic group G is by definition the quotient G/B by a Borel subgroup. It can be regarded as the set of Borel subalgebras of Lie(G). Given a nilpotent element e in Lie(G), one calls Springer fiber the subvariety formed by the Borel subalgebras which contain e. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by C. De Concini, G. Lusztig, and C. Procesi asserts that, when G is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In th…
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 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…
In math.RT/0201073 we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence; provide a similar ``Langlands dual'' description for the category of equivariant coherent sheaves on the nilpotent cone; and deduce some conjectures by Lusztig and Ostrik.
The main result of this paper is a new parameterization of the orbits of a symmetric subgroup on a partial flag variety. The parameterization is in terms of certain Spaltenstein varieties, on one hand, and certain nilpotent orbits, on the other. One of the motivations is related to enumerating special unipotent representations of real reductive groups. Another motivation is understanding (a portion of) the closure order on the set of nilpotent coadjoint orbits.
Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A.
Our goal is to classify all generically transitive actions of commutative unipotent groups on flag varieties up to conjugation. We establish relationship between this problem and classification of multiplications with certain properties on Lie algebra representations. Then we classify multiplications with the desired properties and solve the initial classification problem.
In this paper, we study the set of $Δ$-filtered modules of quasi-hereditary algebras arising from quotients of the double of quivers of type $A$. Our main result is that for any fixed $Δ$-dimension vector, there is a unique (up to isomorphism) exceptional $Δ$-filtered module. We then apply this result to show that there is always an open adjoint orbit in the nilpotent radical of a seaweed Lie algebra in $\mathrm{gl}_{n}(\field)$, thus answering positively in this $\mathrm{gl}_{n}(\field)$ case to a question raised independently by Michel Duflo and Dmitri Panyushev. An example of a seaweed Lie …
We continue the study of the closures of $GL(V)$-orbits in the enhanced nilpotent cone $V\times\cN$ begun by the first two authors. We prove that each closure is an invariant-theoretic quotient of a suitably-defined enhanced quiver variety. We conjecture, and prove in special cases, that these enhanced quiver varieties are normal complete intersections, implying that the enhanced nilpotent orbit closures are also normal.
For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimize the function are representations of $\mbox{A}_1$. For twisted Higgs bundles on the projective line, the quiver describing the bottom of the cone is more complicated. We determine it here. We show that the moduli space is topologically connected whenever the rank and degree are…
We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…
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 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.
We prove that generic higher Deligne-Lusztig representations over truncated formal power series are non-nilpotent, when the parameters are non-trivial on the biggest reduction kernel of the centre; we also establish a relation between the orbits of higher Deligne-Lusztig representations of $\mathrm{SL}_n$ and of $\mathrm{GL}_n$. Then we introduce a combinatorial analogue of Deligne-Lusztig construction for general and special linear groups over local rings; this construction generalises the higher Deligne--Lusztig representations and affords all the nilpotent orbit representations, and for $\m…
In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand-Zeitlin integrable system for the Lie algebra $\fgl(n+1,\C)$ and introduced the strongly regular elements, which are the points where the Gelfand-Zeitlin flow is Lagrangian. Later Colarusso studied the nilfibre, which consists of strongly regular elements such that each $i\times i$ submatrix in the upper left corner is nilpotent. In this paper, we prove that every Borel subalgebra contains strongly regular elements and determine the Borel subalgebras containing elements of the nilfibre by using the theory of $K_{i}=GL(i…
In this paper we give a direct proof of the positivity conjecture for adapted quantum cluster variables. Moreover, our process allows one to explicitly compute formulas for all adapted cluster monomials and certain ordered products of adapted cluster monomials. In particular, we describe all cluster monomials in cluster algebras and quantum cluster algebras of rank 2. One may obtain similar formulas for all finite type cluster monomials. The above results are achieved by computing explicit set-theoretic decompositions of Grassmannians of subrepresentations in adapted valued quiver representati…
It is a remarkable theorem by Maffei--Nakajima that the Slodowy variety, which is a subvariety of the resolution of the nilpotent cone, can be realized as a Nakajima quiver variety of type A. However, the isomorphism is rather implicit as it takes to solve a system of equations in which the variables are linear maps. In this paper, we construct solutions to this system under certain assumptions. This establishes an explicit and efficient way to compute the image of a complete flag contained in the Slodowy variety under the Maffei--Nakajima isomorphism and describe these flags in terms of quive…
By using perverse sheaves on representation spaces of quivers over $k[t]/(t^n)$ and jet schemes over flag varieties, we construct a geometric composition algebra $\mathbf K$ under Lusztig's framework on geometric realizations of the negative part of quantum algebras. Simple perverse sheaves in $\mathbf K$ form the canonical basis of $\mathbf K$. The relationships among the algebra $\mathbf K$, the composition algebra of locally projective representations of quivers over $k[t]/(t^n)$ and quantum generalized Kac-Moody algebra are provided.
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.
We continue, generalize and expand our study of linear degenerations of flag varieties from [G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, Math. Z. 287 (2017), no. 1-2, 615-654]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degener…
Let $\Aa_t$ be the directed quiver of type $\Aa$ with $t$ vertices. For each dimension vector $d$ there is a dense orbit in the corresponding representation space. The principal aim of this note is to use just rank conditions to define the irreducible components in the complement of the dense orbit. Then we compare this result with already existing ones by Knight and Zelevinsky, and by Ringel. Moreover, we compare with the fan associated to the quiver $\Aa$ and derive a new formula for the number of orbits using nilpotent classes. In the complement of the dense orbit we determine the irreducib…
Let $\mathfrak{g}$ be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism $\overline{D}$ on the coordinate ring $\mathbb{C}[N]$ related to Brion's equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram's strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when $\mathfrak{g}$ is of type $A_n$ or $…
We define a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. We study commutativity, cancellativity and associativity of this operation. We focus on rigid irreducible components and discuss inductive ways to construct them.
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.
In this article we define a generalization of Lusztig Lagrangian varieties in the case of arbitrary quivers, possibly carrying loops. As opposed to the Lagrangian varieties constructed by Lusztig, which consisted in nilpotent representations, we have to consider here slightly more general representations. That this is necessary is already clear from the Jordan quiver case. Our proof of the Lagrangian character is based on induction, but with non trivial first steps, consisting in the study of quivers with one vertex but possible loops. From our proof emerges a new combinatorial structure on th…
We compute the Euler characteristics of quiver Grassmannians and quiver flag varieties of tree and band modules and prove their positivity. This generalizes some results by G.C. Irelli [arXiv:0910.2592]. As an application we consider the Ringel-Hall algebra $C(A)$ of some string algebras $A$ and compute in combinatorial terms the products of arbitrary functions in $C(A)$.
A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector space at each vertex. Generically among all nilpotent endomorphisms, there is a well-defined Jordan form for these linear transformations, which is an interesting new invariant of a quiver representation. If $Q$ is a Dynkin quiver and $m$ is a minuscule vertex, we show that representations consisting of direct sums of indecomposable representations all including $m$ in their support, the category of which we denote by $\mathcal{C}_{Q,m}$, are determined up to isomorphism by this invariant. We use …
In previous work we equipped quiver Grassmannians for nilpotent representations of the equioriented cycle with an action of an algebraic torus. We show here that the equivariant cohomology ring is acted upon by a product of symmetric groups and we investigate this permutation action via GKM techniques. In the case of (type A) flag varieties, or Schubert varieties therein, we recover Tymoczko's results on permutation representations.
We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$ on the variety of $x$-nilpotent complex matrices and translate it to a representation-theoretic context. We obtain a criterion as to whether the action admits a finite number of orbits and specify a system of representatives for the orbits in the finite case of $2$-nilpotent matrices. Furthermore, we give a set-theoretic description of their closures and specify the minimal degenerations in detail for the action of the Borel subgroup. We show that in all non-finite cases, the correspon…
Let $Q$ be a finite type quiver i.e. ADE Dynkin quiver. Denote by $Λ$ its preprojective algebra. It is known that there are finitely many indecomposable $Λ$-modules if and only if $Q$ is of type $A_1,A_2,A_3,A_4$. In this paper, extending Lusztig's construction of $U\frak{n}_+$, we study an algebra generated by these indecomposable submodules. It turns out that it forms the universal enveloping algebra of some nilpotent Lie algebra inside the function algebra on Lusztig's nilpotent scheme. The defining relations of the corresponding nilpotent Lie algebra for type $A_1, A_2,A_3,A_4$ are given h…
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…
We give a presentation of localized affine and degenerate affine Hecke algebras of arbitrary type in terms of weights of the polynomial subalgebra and varied Demazure-BGG type operators. We offer a definition of a graded algebra $\mathcal{H}$ whose category of finite-dimensional ungraded nilpotent modules is equivalent to the category of finite-dimensional modules over an associated degenerate affine Hecke algebra. Moreover, unlike the traditional grading on degenerate affine Hecke algebras, this grading factors through central characters, and thus gives a grading to the irreducible representa…
Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac-Moody algebra. The underlying set consists of the irreducible components of Lusztig's quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac-Moody algebras by replacing Lusztig's preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.
