Bernard Leclerc
We introduce some new Frobenius subcategories of the module category of a preprojective algebra of Dynkin type, and we show that they have a cluster structure in the sense of Buan-Iyama-Reiten-Scott. These categorical cluster structures yield cluster algebra structures in the coordinate rings of intersections of opposed Schubert cells.
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We show that the zip stratification given by an arbitrary $\Ghat$-zip over a scheme is pure. We deduce purity of the level-$m$-stratification for truncated Barsotti-Tate groups and purity of the Ekedahl-Oort stratification for special fibers of good models of Shimura varieties of Hodge type. We also show that all Ekedahl-Oort strata are quasi-affine schemes and generalize several properties of the Bruhat stratification from the PEL case to the case of Shimura varieties of Hodge type.
The cohomology of certain intermediate strata of Kottwitz varieties
本文利用迹公式和p-进一般线性群表示，推导出Kottwitz簇某些层调的étale上同调Frobenius-Hecke迹的显式公式。
Linear degenerate symplectic flag varieties: symmetric degenerations and PBW locus
本文在A型等向箭图框架下将线性退化辛旗流形概念化为对称退化，证明不同退化序的等价性，并通过多视角实现研究PBW轨迹中退化簇的几何性质。
We show that in type A or C any degenerate flag variety is in fact isomorphic to a Schubert variety in an appropriate partial flag manifold.
$G$-stable pieces and partial flag varieties
我们将使用$G$ -稳定片的组合学来描述\cite [第4节]{L3}中部分旗簇划分的封闭关系。
Dynkin abelianisations of flag varieties
本文引入Dynkin锥参数化经典李代数的阿贝尔化，将Cerulli Irelli和Lanini关于A、C型旗簇PBW退化为Schubert簇的结果推广到所有源自Dynkin锥的度向量退化情形，并计算了所有经典类型中退化单模的定义关系。
We further develop the general theory of the "mixed modular derived category" introduced by the authors in a previous paper in this series. We then use it to study positivity and Q-Koszulity phenomena on flag varieties.
Equivariant Sheaves on Flag Varieties
证明了复约化群的旗簇上的层的Borel-等变导出范畴等价于单等变反常层的直和的扩张代数上的dg模的完全导出范畴.这证明了Soergel和Lunts在旗簇的情况下的一个猜想。
We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks $G$-Zip$^μ$. Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl-Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags $G$-ZipFlag$^μ$, fibered in flag varieties over $G$-Zip$^μ$. It provides a simultaneous generalization of the "classical case" homogeneous complex manifolds studied by Griffiths-Schmid and the "flag space" for Siegel varieties studied by Ekedahl-van der Geer. Four …
We prove a new family of total positivity criteria for partial flag varieties for simply-connected complex algebraic group in the simply laced case.
Let G be a reductive algebraic group over the complex number filed, and K = G^θ be the fixed points of an involutive automorphism θof G so that (G, K) is a symmetric pair. We take parabolic subgroups P and Q of G and K respectively and consider a product of partial flag varieties G/P and K/Q with diagonal K-action. The double flag variety G/P \times K/Q thus obtained is said to be of finite type if there are finitely many K-orbits on it. A triple flag variety G/P^1 \times G/P^2 \times G/P^3 is a special case of our double flag varieties, and there are many interesting works on the triple flag …
Quivers of type A, flag varieties and representation theory
关于标题所列主题的简短非正式调查。在环与代数国际会议X.
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.
For a given flag variety, we characterize the primes $p$ for which there exists a weight $λ$ such that the Hard Lefschetz Theorem holds for multiplication by $λ$ on the cohomology of the flag variety with coefficients in an infinite field of characteristic $p$.
Tautological rings of Shimura varieties and cycle classes of Ekedahl-Oort strata
该研究定义了志村簇的tautological环，阐明Hodge型志村簇良好约化特殊纤维上该环的结构并证明其由Ekedahl-Oort层闭链类生成，还计算了这些闭链类并给出三项相关应用。
Let $G$ be the split orthogonal group of degree $2n$ over an arbitrary infinite field $\mathbb{F}$ of chararcteristic not $2$. In this paper, we classify multiple flag varieties $G/P_1\times\cdots\times G/P_k$ of finite type. Here a multiple flag variety is said to be of finite type if it has a finite number of $G$-orbits with respect to the diagonal action of $G$.
Affine flag varieties of type D
本文建立仿射D型完全与部分旗簇的格表示，确定其完全旗簇对应的卷积代数同构于扩张仿射Hecke代数，证明该代数存在单项式基与典范基并给出乘法正性性质，拓展了仿射旗簇的几何表示理论。
We construct a monoidal category $\mathscr{C}_{w,v}$ which categorifies the doubly-invariant algebra $^{N'(w)}\mathbb{C}[N]^{N(v)}$ associated with Weyl group elements $w$ and $v$. It gives, after a localization, the coordinate algebra $\mathbb{C}[\mathcal{R}_{w,v}]$ of the open Richardson variety associated with $w$ and $v$. The category $\mathscr{C}_{w,v}$ is realized as a subcategory of the graded module category of a quiver Hecke algebra $R$. When $v= \mathrm{id}$, $\mathscr{C}_{w,v}$ is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent c…
Building on the theory of parity sheaves due to Juteau-Mautner-Williamson, we develop a formalism of "mixed modular perverse sheaves" for varieties equipped with a stratification by affine spaces. We then give two applications: (1) a "Koszul-type" derived equivalence relating a given flag variety to the Langlands dual flag variety, and (2) a formality theorem for the modular derived category of a flag variety (extending a previous result of Riche-Soergel-Williamson).
Twisted loop groups and their affine flag varieties
本文发展了Laurent幂级数局部域k（（t））上约化群的仿射旗簇及其Schubert簇的理论，其中k是完满域.这可以被看作是一个推广的理论，仿射旗品种循环组的“扭曲的情况下”，我们的结果的一个后果是，我们的建设也包括旗品种的卡茨-穆迪李代数的仿射型。给出了k（（t））上固定群上的部分旗簇上的自然充裕线丛的整体瓣空间的维数的一个凝聚猜想，以及在Shimura簇的局部模型中的一些应用.
Let $H$ be a connected spherical subgroup of a semisimple algebraic group $G$. In this paper, we give a criterion for $H$-orbit closures in the flag variety of $G$ to have nice geometric and cohomological properties. Our main tool is the method of Frobenius splitting and of global F-regularity.
We introduce the notion of flag Bott-Samelson variety as a generalization of Bott-Samelson variety and flag variety. Using a birational morphism from an appropriate Bott-Samelson variety to a flag Bott-Samelson variety, we compute Newton-Okounkov bodies of flag Bott-Samelson varieties as generalized string polytopes, which are applied to give polyhedral expressions for irreducible decompositions of tensor products of $G$-modules. Furthermore, we show that flag Bott-Samelson varieties are degenerated into flag Bott manifolds with higher rank torus actions, and find the Duistermaat-Heckman measu…
We determine the set of supports for the flat family of linear degenerations of flag varieties in terms of Motzkin combinatorics.
We describe the effect of Feigin's flat degeneration of the type $\textrm{A}$ flag variety on its Schubert varieties. In particular, we study when they stay irreducible and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some Schubert varieties with Richardson varieties in higher rank partial flag varieties.
We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank. We deduce that the corresponding degenerate flag varieties are isomorphic to Schubert varieties in any characteristic.
Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper arXiv:0805.0045v4 by Haines, Kottwitz, Reuman, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic $σ$-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic $σ$-conjugacy class is the class of $b=1$, we also prove the dimension formula predicted in op. cit. in almost all cases.
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures concerning their dimensions, and generalizes the superset method.
On the Classification of Schubert Varieties in Partial Flag Varieties
本文在部分旗流形中推广了Schubert流形的分类，通过根系折叠得到等价对，并确定了Schubert三重流形等价类的上界。
Regular orbits of symmetric subgroups on partial flag varieties
本文的主要结果是部分旗簇上对称子群轨道的一个新的参数化。参数化一方面是根据某些Spalantis变种，另一方面是根据某些幂零轨道。其中一个动机与列举实约化群的特殊幂幺表示有关。另一个动机是理解幂零余伴轨道集合上的闭包序（的一部分）。
Staggered sheaves on partial flag varieties
交错$t$ -结构是等变相干层的导出范畴上的一类$t$ -结构。本文证明了博雷尔子群等变的部分旗簇上相干层的导出范畴具有Artin交错$t$ -结构。因此，我们得到了一个基础，其等变$K$ -理论组成的简单交错层。
The goal of this paper is to study the link between the topology of the degenerate flag varieties and combinatorics of the Dellac configurations. We define three new classes of algebraic varieties closely related to the degenerate flag varieties of types A and C. The construction is based on the quiver Grassmannian realization of the degenerate flag varieties and odd symplectic and odd and even orthogonal groups. We study basic properties of our varieties; in particular, we construct cellular decomposition in all the three cases above (as well as in the case of even symplectic group). We also …
Affine Pavings of Quiver Flag Varieties
本文为 Dynkin 型箭图偏旗簇构造了仿射铺砌，扩展了多位学者的方法与 Auslander-Reiten 理论技术以实现该结果。
The Jordan-Hoelder series for nearby cycles on some Shimura varieties and affine flag varieties
我们研究的约旦-Hoelder系列附近的周期上某些志村品种和Rapoport-津克本地模型，并在有限维件贝林森的变形的仿射格拉斯曼的仿射旗品种（和他们的p-adic类似物）。本文给出了不可约组分在某些上同调群中重数的计算公式，并给出了在仿射Hecke代数中计算重数的算法.
Radon transforms of twisted D-modules on partial flag varieties
本文研究部分旗簇上扭曲D-模的拉东变换（交织函子）与半单李代数表示的关系，证明其导出范畴等价性（推广Marastoni结果）及与整体截面的兼容性。
We study algebraic, combinatorial and geometric aspects of weighted PBW-type degenerations of (partial) flag varieties in type $A$. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.
For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X.
For a classical simple algebraic group $G$ we obtain the affirmative answer for the conjecture in [8] that there exists an isomorphism between the geometric crystal on the flag variety and the one on the unipotent subgroup $U^-$.
Independence of the Grothendieck classes of twisted symplectic flag varieties
本文利用非交换动机理论，证明扭曲辛旗簇的格罗滕迪克类在线性及齐次二次意义下，在代数簇的格罗滕迪克环中具有独立性。
Recently E. Feigin introduced the $\mathbb G_a^N$-degenerations of semisimple algebraic groups and their associated degenerate flag varieties. It has been shown by Feigin, Finkelberg, and Littelmann that the degenerate flag varieties in types $A_n$ and $C_n$ are Frobenius split. In this paper we construct an associated degeneration of homogeneous coordinate rings of classical flag varieties in all types and show that these rings are Frobenius split in most types. It follows that the degenerate flag varieties of types $A_n$, $C_n$ and $G_2$ are Frobenius split. In particular we obtain an altern…
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of GL_2. At first we determine all such varieties up to isomorphy. After this we investigate the representations of the sigma-stabilizer of an element b of the group on the etale cohomology of the affine Deligne-Lusztig variety X_w(b). We describe such representations as inductions from compact subgroups and in terms of noncuspidal representations.
Let $G$ be a simple algebraic group of type $B_2$ over an algebraically closed field of odd characteristic. We prove that the flag variety $G/B$ is D-affine. This extends an earlier result of H.H.Andersen and M.Kaneda.
Following the historical track in pursuing $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation theory, such as canonical bases, Newton-Okounkov bodies, PBW-filtrations and cluster algebras come to push the subject forward.
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.
Let $\Fl^a_\la$ be the PBW degeneration of the flag varieties of type $A_{n-1}$. These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$. We prove that $\Fl^a_\la$ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\la$ of $\Fl^a_\la$. The varieties $R_\la$ can be viewed as towers of successive $\bP^1$-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties $R_\la$ are Frobenius split. This gives us Frobenius splitting for the…
Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open $G_{\mathbb{R}}$--orbits in flag varieties $G/P$. We investigate Hodge--theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, we relate the Griffiths--Yukawa coupling to the variety of lines on $G/P$ (under a minimal homogeneous embedding), construct a large class of polarized $G_{\mathbb{R}}$--orbits in $G/P$, and compute the associated Hodge--theoretic boundary components. An emphasis is placed through…
Cohomology of Flag Superschemes and Syzygies of Compositional Varieties
研究部分旗超概形的凝聚上同调，计算四类新无限族旗超概形的结构层上同调，引入复合簇并计算其Tor群，证明旗超概形上同调与普通旗簇上同调环及复合簇Tor群的张量积同构。
Tropical symplectic flag varieties: a Lie-theoretic approach
本文用李理论方法研究辛旗簇的热带化，显式刻画了一个极大素锥，并证明其内点对应的格罗布纳退化为FFLV多胞体相关的环面簇，关键工具是双有理序列。
A partial flag variety ${\mathcal {P}}_K$ of a Kac-Moody group $G$ has a natural stratification into projected Richardson varieties. When $G$ is a connected reductive group, a Bruhat atlas for ${\mathcal {P}}_K$ was constructed by He, Knutson and Lu: ${\mathcal {P}}_K$ is locally modeled with Schubert varieties in some Kac-Moody flag variety as stratified spaces. The existence of Bruaht atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties…
We study the $S_n$-equivariant log-concavity of the cohomology of flag varieties, also known as the coinvariant ring of $S_n$. Using the theory of representation stability, we give computer-assisted proofs of the equivariant log-concavity in low degrees and high degrees and conjecture that it holds for all degrees. Furthermore, we make a stronger unimodal conjecture which implies the equivariant log-concavity.
Newton--Okounkov bodies of partial flag varieties via cluster algebras
利用单胞上的簇结构构造部分旗簇中Schubert变体的Newton-Okounkov多面体，证明在无限型簇代数下存在无穷多互不等价的多面体，并应用于构造无穷多不同的单调拉格朗日环面。
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, …
Semiorthogonal decompositions for families of twisted flag varieties
该研究将Samokhin-van der Kallen的半正交分解推广至拟分裂半单群，证明光滑扭曲广义旗簇族的导出范畴可半正交分解，实现了Panin关于扭曲旗簇Quillen K-理论计算的范畴化。
We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by Gross-Hacking-Keel-Kontsevich: for the flag variety the cone is the tropicalization of their superpotential while for Schubert varieties a restriction of the superpotential is necessary. We prove…
We compute toric degenerations arising from the tropicalization of the full flag varieties $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ embedded in a product of Grassmannians. For $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ we compare toric degenerations arising from string polytopes and the FFLV polytope with those obtained from the tropicalization of the flag varieties. We also present a general procedure to find toric degenerations in the cases where the initial ideal arising from a cone of the tropicalization of a variety is not prime.
Gamma conjecture I for flag varieties
本文通过Galkin和Iritani提出的方法，证明了所有旗种的Gamma猜想I，核心在于展示Rietsch镜像的完全正部分与Γ-类的镜像关系，并包含超势能临界点。
On the Smooth Locus in Flat Linear Degenerations of Flag Varieties
本文利用矩图技术给出旗簇平坦线性退化中光滑轨迹的组合刻画，推广了Feigin退化的相关结果，并提出用传递竞赛图判定固定点光滑性的新组合判据。
Let $\SF^a_λ$ be the degenerate symplectic flag variety. These are projective singular irreducible $\bG_a^M$ degenerations of the classical flag varieties for symplectic group $Sp_{2n}$. We give an explicit construction for the varieties $\SF^a_λ$ and construct their desingularizations, similar to the Bott-Samelson resolutions in the classical case. We prove that $\SF^a_\la$ are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel-Weil-Bott theo…
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…
We study the $\bG_a^M$ degenerations $\Fl^a_\la$ of the type $A$ flag varieties $\Fl_\la$. We describe these degenerations explicitly as subvarieties in the products of Grassmanians. We construct cell decompositions of $\Fl^a_\la$ and show that for complete flags the number of cells is equal to the normalized median Genocchi numbers $h_n$. This leads to a new combinatorial definition of the numbers $h_n$. We also compute the Poincar\' e polynomials of the complete degenerate flag varieties via a natural statistics on the set of Dellac's configurations, similar to the length statistics on the s…
We prove sign-alternation of the structure constants in the basis of structure sheaves of opposite Schubert varieties in the torus-equivariant Grothendieck group of coherent sheaves on the flag varieties $G/P$ associated to an arbitrary symmetrizable Kac-Moody group $G$, where $P$ is any parabolic subgroup. This generalizes the work of Anderson-Griffeth-Miller from the finite case to the general Kac-Moody case, and affirmatively answers a conjecture of Lam-Schilling-Shimozono regarding the signs of the structure constants in the case of the affine Grassmannian.
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