G. Lusztig
For any semifield K we define a K-form of a partial flag manifold of a semisimple group G of simply laced type over the complex numbers. The definition is in terms of the theory of canonical bases.
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Let G be a semisimple simply connected group of simply laced type over C. We show that the flag manifold of G has a version defined over the "tropical" semifield Z on which the monoid G(Z) associated to G and the semifield Z acts.
A flag manifold over a semifield K can be partitioned into "half i-circles" which are orbits of a K-action on that flag manifold. Here i is fixed and it corresponds to a simple reflection in the Weyl group. We prove (for certain K) a connectedness property of the flag manifold: any two points of it can be joined by a finite sequence of half i-circles for various i.
The quantum group $\dot U$ and flag manifolds over the semifield Z
本文利用半域 Z 上旗流形，对修正量子群的典范基给出参数化，并修正了此前工作中的一些小错误。
In this paper, we develop the theory of flag manifold over a semifield for any Kac-Moody root datum. We show that the flag manifold over a semifield admits a natural action of the monoid over that semifield associated with the Kac-Moody datum and admits a cellular decomposition. This extends the previous work of Lusztig, Postnikov, Rietsch and others on the totally nonnegative flag manifolds (of finite type) and the work of Lusztig, Speyer, Williams on the tropical flag manifolds (of finite type). As a by-product, we prove a conjecture of Lusztig on the duality of totally nonnegative flag mani…
We will use the combinatorics of the $G$-stable pieces to describe the closure relation of the partition of partial flag varieties in \cite[section 4]{L3}.
We introduce a class of perverse sheaves on a partial flag manifold of a connected reductive group G defined over a finite field which are equivariant under the action of the group of rational points of G. The definition of this class is similar to the definition of parabolic character sheaves.
Staggered $t$-structures are a class of $t$-structures on derived categories of equivariant coherent sheaves. In this note, we show that the derived category of coherent sheaves on a partial flag variety, equivariant for a Borel subgroup, admits an artinian staggered $t$-structure. As a consequence, we obtain a basis for its equivariant $K$-theory consisting of simple staggered sheaves.
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.
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。
A geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g. a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous vector bundle over homogeneous spaces. In this paper, we study the relationship between partial flag and complete flag cases. Our main result is that the locally finite modules over the geometric extension algebras are related by a recollement. As examples, we investigate parabolic affine nil Hecke algebras, geometric extension algebras associated to parabolic Sp…
Cohomological induction gives an algebraic method for constructing representations of a real reductive Lie group $G$ from irreducible representations of reductive subgroups. Beilinson-Bernstein localization alternatively gives a geometric method for constructing Harish-Chandra modules for $G$ from certain representations of a Cartan subgroup. The duality theorem of Hecht, Mili\vcić, Schmid and Wolf establishes a relationship between modules cohomologically induced from minimal parabolics and the cohomology of the $\ms{D}$-modules on the complex flag variety for $G$ determined by the Beilinson-…
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。
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…
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 …
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.
We describe a categorification of the cluster algebra structure of multi-homogeneous coordinate rings of partial flag varieties of arbitrary Dynkin type using Cohen-Macaulay modules over orders. This completes the categorification of Geiss-Leclerc-Schröer by adding the missing coefficients. To achieve this, for an order $A$ and an idempotent $e \in A$, we introduce a subcategory $\operatorname{CM}\nolimits_e A$ of $\operatorname{CM}\nolimits A$ and study its properties. In particular, under some mild assumptions, we construct an equivalence of exact categories $(\operatorname{CM}\nolimits_e A)…
We prove a new family of total positivity criteria for partial flag varieties for simply-connected complex algebraic group in the simply laced case.
An explicit formula for the Hilbert series of a partial flag variety
本文利用等变嵌入的希尔伯特多项式等于希尔伯特函数，给出了部分旗簇希尔伯特级数的显式公式，并以$SL(n+1,\mathbb{C})$的伴随簇为例，计算了$A$型基本表示分子线性项。
Half-integrality of line bundles on partial flag schemes of classical Lie groups
本文发展部分旗概形上等变线丛的 Galois 下降理论，考察抛物子群特征所附下降数据分类的计算问题，并据此分类经典李群标准 $\mathbb{Z}[1/2]$-形式上的等变线丛。
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。
Let L be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories sub(Q) for Q an injective L-module, and we introduce a mutation operation between complete rigid modules in sub(Q). This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.
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 $ G $ be a connected reductive algebraic group over $ \C $. We denote by $ K = (G^θ)_{0} $ the identity component of the fixed points of an involutive automorphism $ θ$ of $ G $. The pair $ (G, K) $ is called a symmetric pair. Let $Q$ be a parabolic subgroup of $K$. We want to find a pair of parabolic subgroups $P_{1}$, $P_{2}$ of $G$ such that (i) $P_{1} \cap P_{2} = Q$ and (ii) $P_{1} P_{2}$ is dense in $G$. The main result of this article states that, for a simple group $G$, we can find such a pair if and only if $(G, K)$ is a Hermitian symmetric pair. The conditions (i) and (ii) yield …
In this paper, we give an explicit formula of Chevalley type, in terms of the Bruhat graph, for the quantum multiplication with the class of the line bundle associated to the anti-dominant minuscule fundamental weight $- \varpi_{k}$ in the torus-equivariant quantum $K$-group of the partial flag manifold $G/P_{J}$ (where $J = I \setminus \{k\}$) corresponding to the maximal (standard) parabolic subgroup $P_{J}$ of minuscule type in type $A$, $D$, $E$, or $B$. This result is obtained by proving a similar formula in a torus-equivariant $K$-group of the semi-infinite partial flag manifold $\mathbf…
Failure of Bott vanishing for (co)adjoint partial flag varieties
该研究证明了（co）伴随偏旗 manifold 不满足 Bott 消失现象，验证了相关猜想。
Assume that the valuation semigroup $Γ(λ)$ of an arbitrary partial flag variety corresponding to the line bundle $\mathcal L_λ$ constructed via a full-rank valuation is finitely generated and saturated. We use Ehrhart theory to prove that the associated Newton-Okounkov body, which happens to be a rational, convex polytope, contains exactly one lattice point in its interior if and only if $\mathcal L_λ$ is the anticanonical line bundle. Furthermore we use this unique lattice point to construct the dual polytope of the Newton-Okounkov body and prove that this dual is a lattice polytope using a r…
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。
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。
We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a gl_n partial flag variety. These hypergeometric solutions manifest the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety.
We study the level $m$--Demazure flag of a level $\ell$--Demazure module for $\frak{sl}_2[t]$. We define the generating series $A_n^{\ell \rightarrow m}(x,q)$ which encodes the $q$--multiplicity of the level $m$ Demazure module of weight $n$. We establish two recursive formulae for these functions. We show that the specialization to $q=1$ is a rational function involving the Chebyshev polynomials. We give a closed form for $A_n^{\ell \rightarrow \ell+1}(x,q)$ and prove that it is given by a rational function. In the case when $m=\ell+1$ and $\ell=1,2$, we relate the generating series to partia…
The main goal of this paper is to show that the (multi-homogeneous) coordinate ring of a partial flag variety $\mathbb{C} [G / P_K^{-}]$ admits a cluster algebra structure if $G$ is any simply-connected semisimple complex algebraic group. We use derivation properties and a special lifting map to prove that the cluster algebra structure $\mathcal{A}$ of the coordinate ring $\mathbb{C}[N_K]$ of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure $\hat{\mathcal{A}}$ living in the coordinate ring of the corresponding partial flag variety. T…
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 equivariant contact structures on complex projective varieties arising as partial flag varieties $G/P$, where $G$ is a connected, simply-connected complex simple group of type $ADE$ and $P$ is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these types. The result can be deduced from Boothby's classification of compact simply-connected complex contact manifolds with transitive action by contact automorphisms, but our proof is completely independent and relies on properties of $G$-equivariant vector bundles on $G/P$. A byprod…
On two notions of total positivity for generalized partial flag varieties of classical Lie types
本文刻画经典李型广义部分旗簇中 Lusztig 全正性与 Plücker 正性相一致的情形，将 Bloch–Karp 的 A 型结果推广到 C 型和 B 型。
On quantum $K$-groups of partial flag manifolds
本文证明部分旗流形的等变小量子$K$-群是全旗流形量子$K$-群的商，尊重Schubert基，是Peterson定理抛物版本的$K$-理论类比，并延续了作者先前工作。
Toda-Type Presentations for the Quantum K Theory of Partial Flag Varieties
本文通过推前技术为部分旗种的量子K理论提供了一种不依赖量子参数的递归公式。
Equivariant K-theory of the space of partial flags
本文通过Drinfeld生成元定义$q=0$仿射量子群代数$\mathfrak{U}_n$，并利用部分旗簇的等变K理论卷积积构造仿射0-Schur代数，证明两者间存在满同态。
In this paper, we solve the initial value problems of variable-coefficient generalized wave equations associated with trees and a large family of linear constant-coefficient partial differential equation by algebraic methods. Moreover, we find all the polynomial solutions for a 3-dimensional variable-coefficient flag partial differential equation of any order, the linear wave equation with dissipation and the generalized anisymmetrical Laplace equation. Furthermore, the polynomial-trigonometric solutions of a generalized Klein-Gordan equation associated with 3-dimensional generalized Tricomi o…
We consider the cotangent bundle T^*F_λof a GL_n partial flag variety, λ= (λ_1,...,λ_N), |λ|=\sum_iλ_i=n, and the torus T=(C^*)^{n+1} equivariant cohomology H^*_T(T^*F_λ). In [MO], a Yangian module structure was introduced on \oplus_{|λ|=n} H^*_T(T^*F_λ). We identify this Yangian module structure with the Yangian module structure introduced in [GRTV]. This identifies the operators of quantum multiplication by divisors on H^*_T(T^*F_λ), described in [MO], with the action of the dynamical Hamiltonians from [TV2, MTV1, GRTV]. To construct these identifications we provide a formula for the stable …
Nonexistence of regular maps between homogeneous projective varieties
本文研究部分旗簇间非恒定正则映射的不存在性，证明部分非完全旗簇到完全旗簇不存在此类映射，并提出一般猜想。
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。
Let $G$ be the group scheme $\operatorname{SL}_{d+1}$ over $\mathbb{Z}$ and let $Q$ be the parabolic subgroup scheme corresponding to the simple roots $α_{2},\cdots,α_{d-1}$. Then $G/Q$ is the $\mathbb{Z} $-scheme of partial flags $\{D_{1}\subset H_{d}\subset V\}$. We will calculate the cohomology modules of line bundles over this flag scheme. We will prove that the only non-trivial ones are isomorphic to the kernel or the cokernel of certain matrices with multinomial coefficients.
Quantum K-Rings of Partial Flag Varieties, Coulomb Branches, and the Bethe Ansatz
本文通过纯几何方法解释了 3D GLSM 的 Coulomb 分支方程与 5-顶点格点模型 Bethe Ansatz 方程的巧合，证明了旗流形量子 $K$-环的两种显式表示，并证明了部分旗流形的稳定映射与拟映射 $K$-理论同构。
We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a "twisted case"; a consequence of our results is that our construction also includes the flag varieties for Kac-Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t…
Given integers $1 \le k_1 < \cdots < k_l \le n-1$, let $\text{Fl}_{k_1,\dots,k_l;n}$ denote the type $A$ partial flag variety consisting of all chains of subspaces $(V_{k_1}\subset\cdots\subset V_{k_l})$ inside $\mathbb{R}^n$, where each $V_k$ has dimension $k$. Lusztig (1994, 1998) introduced the totally positive part $\text{Fl}_{k_1,\dots,k_l;n}^{>0}$ as the subset of partial flags which can be represented by a totally positive $n\times n$ matrix, and defined the totally nonnegative part $\text{Fl}_{k_1,\dots,k_l;n}^{\ge 0}$ as the closure of $\text{Fl}_{k_1,\dots,k_l;n}^{>0}$. On the other …
Cohomology on Cotangent Bundles of Partial Flag Varieties in Type A
针对A型特殊线性群的偏Flag簇余切丛上的向量丛，通过与Bott--Samelson簇上同调的比较得到高阶上同调消失等结论，进而给出相关函数环的生成元集并证明其仿射化具有终端奇点。
Partial Hasse invariants for Shimura varieties of Hodge-type
本文在$G$-zip旗栈上定义偏Hasse不变量，并在Hodge型Shimura簇旗空间获得类似截面，证明其作为模$p$自守形式可切出余维1层，且能通过高阶自守向量丛自然分解并位于其socle中。
We continue our study of the inclusion posets of diagonal $SL(n)$-orbit closures in a product of two partial flag varieties. We prove that, if the diagonal action is of complexity one, then the poset is isomorphic to one of the 28 posets that we determine explicitly. Furthermore, our computations show that the number of diagonal $SL(n)$-orbits in any of these posets is at most 10 for any positive integer $n$. This is in contrast with the complexity 0 case, where, in some cases, the resulting posets attain arbitrary heights.
Regularity theorem for totally nonnegative flag varieties
本文证明了偏旗流形$G/P$的全非负部分是正则CW复形，证实了Williams的猜想，并特别证实了全非负Grassmannian中positroid胞腔闭包同胚于球的Postnikov猜想。
Cluster structures on spinor helicity and momentum twistor varieties
本文研究部分旗簇与格拉斯曼流形在普吕克嵌入下的齐次坐标环，证明嵌入保持簇代数结构，并应用于散射振幅中旋量螺旋度与动量扭量簇的簇关系。
The quantum enveloping algebra of $\mathfrak{sl}_n$ (and the quantum Schur algebras) was constructed by Beilinson-Lusztig-MacPherson as the convolution algebra of $GL_d$-invariant functions over the space of pairs of partial $n$-step flags over a finite field. In this paper we expand the construction to the mirabolic setting of triples of two partial flags and a vector, and examine the resulting convolution algebra. In the case of $n=2$, we classify the finite dimensional irreducible representations of the mirabolic quantum algebra and we prove that the category of such representations is semi…
Hochschild cohomology of generalised Grassmannians
本文计算广义格拉斯曼流形的Hochschild上同调的HKR分解，证明其在（余）极小型和（余）伴随型情形集中于整体截面，并给出Gerstenhaber代数结构的显式部分描述。
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