Grassmannians and representations

主引物格拉斯曼流形：猜想

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。

Singularities of the dual varieties associated to exterior representations: 1. Dual Grassmannian

For a given irreducible projective variety $X$, the closure of the set of all hyperplanes containing tangents to $X$ is the projectively dual variety $X^{\vee}$. We study the singular locus of projectively dual varieties of certain Segre-Plücker embeddings in series of papers. In this work we give a classification of the irreducible components of the singular locus of the dual Grassmannian. Basically, it admits two components: cusp type and node type which are degeneracies of a certain Hessian matrix, and the closure of the set of tangent planes having more than one critical point, respectivel…

Rationality of Rigid Quiver Grassmannians

We show that any quiver Grassmannian associated with a rigid representation of a quiver is a rational variety using torus localization techniques.

Modular representations of reductive groups and geometry of affine Grassmannians

By the geometric Satake isomorphism of Mirkovic and Vilonen, decomposition numbers for reductive groups can be interpreted as decomposition numbers for equivariant perverse sheaves on the complex affine Grassmannian of the Langlands dual group. Using a description of the minimal degenerations of the affine Grassmannian obtained by Malkin, Ostrik and Vybornov, we are able to recover geometrically some decomposition numbers for reductive groups. In the other direction, we can use some decomposition numbers for reductive groups to prove geometric results, such as a new proof of non-smoothness res…

主箭图格拉斯曼流形的极值性

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。

Cell decompositions and algebraicity of cohomology for quiver Grassmannians

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…

由单个矩阵定义的幂零循环表示所关联的箭图Grassmannian

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。

弦表示的箭图格拉斯曼流形上的骨架环面作用与GKM结构

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。

迷向Grassmann流形的Hochschild上同调

Hochschild Cohomology of Isotropic Grassmannians

本研究证明非特殊迷向Grassmann流形不具有Hochschild全局性，证实了Belmans与Smirnov的相关猜想，并得出Bott消没对这类簇不成立的推论。

Desingularization of quiver Grassmannians for Dynkin quivers

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.

Dynkin箭图的箭图Grassmannian的旗版本没有奇上同调

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。

The Cohomology of the Grassmannian is a $gl_n$-module

The integral singular cohomology ring of the Grassmann variety parametrizing $r$-dimensional subspaces in the $n$-dimensional complex vector space is naturally an irreducible representation of the Lie algebra of all the $n\times n$ matrices with integral entries. Using the notion of Schubert derivation, a distinguished Hasse-Schmidt derivation on an exterior algebra, we describe explicitly such a representation, indicating its relationship with the celebrated bosonic vertex representation of the Lie algebra of infinite matrices due to Date, Jimbo, Kashiwara and Miwa.

Cell Decompositions for Rank Two Quiver Grassmannians

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…

Factorial Schur and Grothendieck polynomials from Bott Samelson varieties

We use Bott-Samelson resolutions of Schubert varieties in Grassmannians along with equiariant localization techniques to show that the factorial Schur functions and the factorial Grothendieck polynomials represent Schubert classes in equivariant cohomology and equivariant K-theory respectively.

Quiver Grassmannians of extended Dynkin type D - Part 2: Schubert decompositions and F-polynomials

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…

Framed moduli and Grassmannians of submodules

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 $A_{n-1}^{(1)}$.

GKLO表示：类型$\mathsf{AI}$的扭曲Yangian及仿射Grassmannian对称商量子化

GKLO representations of twisted Yangians in type $\mathsf{AI}$ and quantizations of symmetric quotients of the affine Grassmannian

本文构造了类型AI扭曲Yangian的GKLO表示，定义其截断，并证明截断的扭曲Yangian量子化仿射Grassmannian横截切片商的概形。

有界核心划分与博雷-韦-博特定理

Bounded core partitions and Borel-Weil-Bott

本文基于博雷-韦-博特定理，通过分析分解，提出两种计算相关Hodge数的公式，并引入新的整数钩积统计量和半标准 Young表，改进了Hodge数正性的已有界限。

到 Grassmann 簇的双有理映射、表示与偏序集多胞体

Birational maps to Grassmannians, representations and poset polytopes

本文研究射影空间到 Grassmann 簇的双有理映射的图闭包，给出其显式描述、纤维计算及到退化李代数循环表示射影化的嵌入，并利用新偏序集多胞体推广 FFLV 基，得到平坦环状退化。

Quiver Grassmannians of type extended Dynkin type D - Part 1: Schubert systems and decompositions into affine spaces

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…

Kronecker箭图的表示与Grassmann流形上的Steiner丛

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。

Quiver Grassmannians and degenerate flag varieties

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…

仿射Grassmannian中的圈关系及其在G-晶体表示的Breuil--Mézard中的应用

Cycles relations in the affine grassmannian and applications to Breuil--Mézard for G-crystalline representations

本文通过仿射Grassmannian中的闭子概形圈关系实现了$\widehat{G}$-表示的Grothendieck群恒等式，并将其与$G$-值晶体表示模空间几何相联系，推导出小Hodge类型下Breuil--Mézard猜想的一个方向。

$\mathrm{GL}_n$-structure and principal $\mathfrak{sl}_2$-triple on the cohomology ring of complex Grassmannian

In this note we describe the cohomology ring of the Grassmannian of $k$-planes in $n$-dimensional complex vector space as an $\mathrm{GL}_n$-module. We give explicit formulas for the operators of its principal $\mathfrak{sl}_2$-triple. It is proved that one of these operators corresponds to the shifted cohomology degree operator and the second operator coincides with the Lefschetz map on cohomology (as in the hard Lefschetz theorem). We check that the cohomology ring of the complex Grassmannian as a $\mathrm{GL}_n$-representation is isomorphic to the $k$-th exterior power of the standard $n$-d…

整数环上Grassmann簇的导出范畴与模表示论

Derived categories of Grassmannians over integers and modular representation theory

本文在整数环上研究Grassmann簇的导出范畴，证明其具有以GL_k和GL_{n-k}表示导出范畴子范畴为分量的半正交分解，并构造倾斜向量丛，揭示其自同态代数的分裂拟遗传结构，结果可推广到任意交换基环。

广义格拉斯曼流形自映射的分类

A Classification of Self-Maps of Generalized Grassmannians

本文针对李型广义格拉斯曼流形，结合根与权系几何等方法，证明其非零次数自映射的上同调诱导作用仅为亚当斯运算或 Dynkin 对称性两类，完成了此类自映射的分类。

Schubert decompositions for quiver Grassmannians of tree modules

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…

A Gelfand-MacPherson correspondence for quiver moduli

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.

D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras

Let ${\mathfrak g}$ be a simple Lie algebra. For a level $κ$ (thought of as a symmetric ${\mathfrak g}$-invariant form of ${\mathfrak g}$), let $\hat{\mathfrak g}_κ$ be the corresponding affine Kac-Moody algebra. Let $Gr_G$ be the affine Grassmannian of ${\mathfrak g}$, and let $D_κ(Gr_G)-mod$ be the category of $κ$-twisted right D-modules on $Gr_G$. By taking global sections of a D-module, we obtain a functor $Γ:D_κ(Gr_G)-mod\to {\mathfrak g}_κ-mod$. It is known that this functor is exact and faithful when $κ$ is negative or irrational. In this paper, we show that the functor $Γ$ is exact and…

Bott-Chern Forms and Arithmetic Intersections

Let \E : 0 --> S --> E --> Q --> 0 be a short exact sequence of hermitian vector bundles with metrics on S and Q induced from that on E. We compute the Bott-Chern form of \E corresponding to any characteristic class, assuming E is projectively flat. The result is used to obtain a new presentation of the Arakelov Chow ring of the arithmetic grassmannian.

关于格拉斯曼流形量子K理论中的赛德尔表示

On Seidel representation in quantum K-theory of Grassmannians

该研究通过具体分析投影格罗莫夫-维滕簇直接证明了格拉斯曼流形量子K理论中的赛德尔表示，给出了量子皮埃里法则的替代证明，简化了高次量子舒伯特结构常数，并为QK(Gr(3,n))提供了量子利特尔伍德-理查森法则。

超格雷厄姆 manifold 的 Borel-Weil 分解

Borel-Weil factorization for super Grassmannians

本文通过显式多线性操作构造超格雷厄姆 manifold 上施施尔 functors 的上同调，揭示其作为自由模的结构及不可约表示的性质。

On the action of the dual group on the cohomology of perverse sheaves on the affine grassmannian

It was proved by Ginzburg and Mirkovic-Vilonen that the $G(O)$-equivariant perverse sheaves on the affine grassmannian of a connected reductive group $G$ form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group $G^\vee$. The proof use the Tannakian formalism. The purpose of this paper is to construct explicitely the action of $G^\vee$ on the global cohomology of a perverse sheaf.

The real loci of Calogero-Moser spaces, representations of rational Cherednik algebras and the Shapiro conjecture

We prove a criterion for the reality of irreducible representations of the rational Cherednik algebras H_{0,1}(S_n). This is shown to imply a criterion for the real loci of the Calogero-Moser spaces C_n in terms of the Etingof-Ginzburg finite maps Υ\colon C_n \to C^n/S_n \times C^n/S_n, recovering a result of Mikhin, Tarasov, and Varchenko [MTV2]. As a consequence we obtain a criterion for the real locus of the Wilson's adelic Grassmannian of rank one bispectral solutions of the KP hierarchy. Using Wilson's first parametrisation of the adelic Grassmannian, we give a new proof of a result of [M…

格拉斯曼流形 $Gr(k,n)$ 的多项式超势：来自顶点函数的极限

Polynomial superpotential for Grassmannian $Gr(k,n)$ from a limit of vertex function

本文通过余切丛顶点函数的积分表示，在 $\hbar\to\infty$ 极限下得到格拉斯曼流形 $Gr(k,n)$ 的 $A$-系列镜像积分及 Eguchi-Hori-Xiong 超势，并发现 Dwork 型同余。

自对偶格拉斯曼流形、Wronski映射与$\mathfrak{gl}_N$、${\mathfrak{sp}}_{2r}$、${\mathfrak{so}}_{2r+1}$的表示

Self-dual Grassmannian, Wronski map, and representations of $\mathfrak{gl}_N$, ${\mathfrak{sp}}_{2r}$, ${\mathfrak{so}}_{2r+1}$

本文利用$\mathfrak{gl}_N$表示论对格拉斯曼流形进行分层并刻画闭包关系，进而将类似分层推广到自对偶格拉斯曼流形，由辛代数或奇正交代数表示论控制。

Quantum Groups, the loop Grassmannian, and the Springer resolution

We establish equivalences of derived categories of the following 3 categories: (1) Principal block of representations of the quantum at a root of 1; (2) G-equivariant coherent sheaves on the Springer resolution; (3) Perverse sheaves on the loop Grassmannian for the Langlands dual group. The equivalence (1)-(2) is an `enhancement' of the known expression for quantum group cohomology in terms of nilpotent variety, due to Ginzburg-Kumar. The equivalence (2)-(3) is a step towards resolving an old mystery surrounding the existense of two completely different realizations of the affine Hecke algebra…

Perverse Sheaves on Loop Grassmannians and Langlands Duality

We outline a proof of a geometric version of the Satake isomorphism. Given a connected, complex algebraic reductive group G we show that the tensor category of representations of the dual group $\check G$ is naturally equivalent to a certain category of perverse sheaves on the loop Grassmannian of G. The above result has been announced by Ginsburg in \cite{G} and some of the arguments are borrowed from \cite{G}. However, we use a more "natural" commutativity constraint for the convolution product, due to Drinfeld. Secondly, we give a direct geometric proof that the global cohomology functor is…

GrassTop：用于病毒分类和系统发育分析的格拉斯曼k-mer拓扑

GrassTop: Grassmannian k-mer Topology for Viral Classification and Phylogenetic Analysis

GrassTop结合格拉斯曼流形与代数拓扑，通过多尺度k-mer描述符和低秩子空间距离实现病毒分类与系统发育分析，在多个数据集上优于五种无比对方法，UPGMA树标签纯度达1.0。

Vanishing cotangent cohomology for Plücker algebras

We use representation theory and Bott's theorem to show vanishing of higher cotangent cohomology modules for the homogeneous coordinate ring of Grassmannians in the Plücker embedding. As a biproduct we answer a question of Wahl about the cohomology of the square of the ideal sheaf for the case of Plücker relations.