Dosang Joe, Bumsig Kim
We found an explicit description of all $GL(n,\RR)$-Whittaker functions as oscillatory integrals and thus constructed equivariant mirrors of flag manifolds. As a consequence we proved the Virasoro conjecture for flag manifolds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Unfolding of equivariant F-bundles and application to the mirror symmetry of flag varieties
本文建立等变 F-丛的展开定理以推广 Hertling-Manin 的亚纯联络万有展开，并据此由小量子上同调结果推出旗簇大量子上同调的镜像对称定理。
We obtain an explicit presentation of the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation of the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational cobordism rings of wonderful symmetric varieties of minimal rank.
The Leray-Hirsch Theorem for equivariant oriented cohomology of flag varieties
本研究借助形式仿射Demazure代数建立旗簇环面等变定向上同调的显式Leray-Hirsch定理，并将相关Borel模型推广至部分旗簇，拓展了等变上同调理论的适用范围。
Let G be a semisimple affine algebraic group and P a parabolic subgroup of G. We classify all flag varieties G/P which admit an action of the commutative unipotent group G_a^n with an open orbit.
We use geometry to give a short proof of an equivariant Pieri rule in the classical flag manifold. This rule is due to Robinson, who gave an algebraic proof.
We prove that the invariant Hilbert scheme parametrising the equivariant deformations of the affine multicone over a flag variety is, under certain hypotheses, an affine space. The proof is based on the construction of a wonderful variety in a fixed multiprojective space.
We prove an explicit combinatorial formula for certain structure constants of the T-equivariant cohomology of the flag manifold SLn/B. Our result generalizes the Pieri-type formula in ordinary cohomology proved by Sottile in 1996. Our result also gives a Pieri-type formula for the double Schubert polynomials introduced by Lascoux and Schutzenberger.
We obtain A- and B- series of flag manifolds for T-equivariant tangent bundles and equivariant Chern polynomial following the arguments of Lian, Liu, Yau.
We study the equivariant K-group of the affine flag manifold with respect to the Borel group action. We prove that the structure sheaf of the (infinite-dimensional) Schubert variety in the K-group is represented by a unique polynomial, which we call the affine Grothendieck polynomial.
Consider the natural torus action on a partial flag manifold $Fl$. Let $Ω_I\subset Fl$ be an open Schubert variety, and let $c^{sm}(Ω_I)\in H_T^*(Fl)$ be its torus equivariant Chern-Schwartz-MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}(Ω_I)$, as well as a formula, of `localization type', for $c^{sm}(Ω_I)$. In fact, we proved similar results for a class $κ_I\in H_T^*(Fl)$ --- in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{SM}(Ω_I)=κ_I$.
Explicit combinatorial cancellation-free rules are given for the product of an equivariant line bundle class with a Schubert class in the torus-equivariant K-theory of a Kac-Moody flag manifold. The weight of the line bundle may be dominant or antidominant, and the coefficients may be described either by Lakshmibai-Seshadri paths or by the alcove model of the first author and Postnikov. For Lakshmibai-Seshadri paths, our formulas are the Kac-Moody generalizations of results of Griffeth-Ram and Pittie-Ram for finite dimensional flag manifolds. A gap in the proofs of the mentioned results is add…
Borel-type presentation of the torus-equivariant quantum $K$-ring of flag manifolds of type $C$
本文给出了$C$型旗流形的环面等变（小）量子$K$-环的Borel型展示。
Loop structure on equivariant $K$-theory of semi-infinite flag manifolds
本文证明了仿射Grassmannian等变K-群的Pontryagin乘积结构与半无限旗流形等变K-群的张量结构一致，并构造了其与局部化等变量子K-群的显式同构，为理解等变量子K-理论环结构与Peterson同构提供了新框架。
Structure Constants in equivariant oriented cohomology of flag varieties
该研究通过形式仿射Demazure代数的纯代数方法，推导旗簇等变定向上同调中变体Bott-Samelson类的结构常数公式，可复现已有几何框架下的相关经典公式，并得到K-理论稳定基公式。
In this article we obtain many results on the multiplicative structure constants of $T$-equivariant Grothendieck ring of the flag variety $G/B$. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_{T}(G/B)$ to $R(T)\otimes R(T)$, where $R(T)$ denotes the representation ring of the torus $T$. We further apply our results to describe the multiplicative structure constants of $K(X)_{\mathbb Q}$ where $X$ is the wonderful compactification of the adjoint group of $G$, in terms of the structure constants of Schubert varieties in the Grothendieck ring of $G/B$.
We introduce a mutation algorithm for puzzles that is a three-direction analogue of the classical jeu de taquin algorithm for semistandard tableaux. We apply this algorithm to prove our conjectured puzzle formula for the equivariant Schubert structure constants of two-step flag varieties. This formula gives an expression for the structure constants that is positive in the sense of Graham. Thanks to the equivariant version of the `quantum equals classical' result, our formula specializes to a Littlewood-Richardson rule for the equivariant quantum cohomology of Grassmannians.
We describe the torus-equivariant cohomology of weighted partial flag orbifolds ${\mathrm{w}}Σ$ of type $A$. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to as ``Schubert Calculus on ${\mathrm{w}}Σ$''. For the weighed Schubert classes in ${\mathrm{w}}Σ$, we give the Chevalley's formula. In addition, we define the weighted analogue of double Schubert polynomials and give the corresponding Chevalley--Monk's formula.
In this paper, we study equivariant vector bundles on partial flag varieties arising from Schur functors. We show that a partial flag variety with three or more steps does not admit an Ulrich bundle of this form with respect to the minimal ample class. We classify Ulrich bundles of this form on two-step flag varieties F(2,n;n+1), F(2,n;n+2), F(k,k+1;n), and F(k,k+2;n). We give a conjectural description of the two-step flag varieties which admit such Ulrich bundles. Our results provide counterexamples to conjectures by Costa and Miro-Roig.
We introduce a conjecture on Virasoro constraints for the moduli space of stable sheaves on a smooth projective surface. These generalise the Virasoro constraints on the Hilbert scheme of a surface found by Moreira (arXiv:2008.13746) and Moreira, Oblomkov, Okounkov and Pandharipande (arXiv:2008.12514). We verify the conjecture in many nontrivial cases by using a combinatorial description of equivariant sheaves found by Klyachko.
Seidel product formula in equivariant quantum $K$-theory of flag varieties
本文在旗种的等价量子K-理论中证明了Seidel乘积公式，并基于Kato的Peterson同构和Ikeda等人研究的nil-Hecke代数给出了证明。
We construct a basis of the equivariant $K$-theory of Bott towers, and we describe precisely the multiplicative structure of these algebras. We deduce similar results for Bott-Samelson varieties. Thanks to the link between flag varieties and Bott-Samelson varieties, we give a method to compute the structure constants of the equivariant $K$-theory of flag varieties in the basis constructed by Kostant and Kumar.
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…
$T$-equivariant motives of flag varieties
本文利用Khan-Ravi稳定同伦范畴构造，将旗簇的T-等变K-理论环谱分解为以Weyl群为指标的BT的K-理论谱直和，并由此推导出T-等变相交理论与K-理论的经典结果。
A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part I: the defining ideal
本文给出 A 型旗流形环面等变量子 K-理论环的显式表示，通过 Q=0 极限和 Nakayama 型论证及逆 Chevalley 公式证明，区别于非等变情形。
We give explicit formulas for torus-equivariant fundamental classes of closed $K$-orbits on the flag variety $G/B$ when $G$ is one of the classical groups $SL(n,\C)$, $SO(n,\C)$, or $Sp(2n,\C)$, and $K$ is a symmetric subgroup of $G$. We describe parametrizations of each orbit set and the combinatorics of its weak order, allowing us to compute formulas for the equivariant classes of all remaining orbit closures using divided difference operators. In each of the cases in type A, we realize the $K$-orbit closures as universal degeneracy loci of a certain type, involving a vector bundle $V$ over …
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.
We prove some general results on the T-equivariant K-theory K_T(G/P) of the flag variety G/P, where G is a semisimple complex algebraic group, P is a parabolic subgroup and T$ is a maximal torus contained in P. In particular, we make a conjecture about a positivity phenomenon in K_T(G/P) for the product of two basis elements written in terms of the basis of K_T(G/P) given by the dual of the structure sheaf (of Schubert varieties) basis. (For the full flag variety G/B, this dual basis is closely related to the basis given by Kostant-Kumar.) This conjecture is parallel to (but different from) th…
Virasoro Constraints for Toric Bundles
本文证明 Gromov–Witten 理论中的 Virasoro 猜想对环面丛全空间成立当且仅当对底空间成立，通过局部化公式和非等变极限结合 Brown 镜像定理实现。
Action of Weyl group on equivariant K-theory of flag varieties
本文研究半单线性群Weyl群对旗簇等变K-理论环的作用，利用分差算子和局部化定理计算反射自同构公式，并在Schubert簇结构层类基下展开，给出有效近似及motivic Chern类基下的讨论。
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。
We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant K-theory of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for anti-dominant fundamental weights in the (small) torus-equivariant quantum K-theory of the flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of the mentioned quantum K-theory. Moreover, in type A, we prove that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivar…
We give a short, geometric proof of Graham's theorem on positivity in the equivariant cohomology of a flag variety, based on a transversality argument.
Wall-crossing formula and genus-one Virasoro conjecture for Fano complete intersections
该研究在射影空间光滑法诺完全交的ambient状态空间上证明了亏格一Virasoro猜想，推广了拟映射不变量的跨壁公式，为完全交与射影空间的Gromov-Witten理论建立了关键桥梁。
We show a Z^2-filtered algebraic structure and a "quantum to classical" principle on the torus-equivariant quantum cohomology of a complete flag variety of general Lie type, generalizing earlier works of Leung and the second author. We also provide various applications on equivariant quantum Schubert calculus, including an equivariant quantum Pieri rule for any partial flag variety of Lie type A.
We use the fixed point arrangement technique developed by Goresky-MacPherson to calculate the part of the equivariant cohomology of the affine flag varieties generated by degree 2. This turns out to be a quadric cone. We also describe the spectrum of the full equivariant cohomology ring as an explicit geometric object. We use our results to show that the vertices of the moment map images of the affine flag varieties lie on a paraboloid.
Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexification. A special class of flag domains constitute the classifying spaces for variations of Hodge structure, namely period domains or more generally Mumford-Tate domains. In this paper we classify all proper, equivariant embeddings of a flag domain of Hodge type, required to satisfy a certain transversality condition, in a period domain and show that any such flag domain must be a Hermitian symmetric space of non-compact type of classical type. In particular, the classification describes also the pr…
Residue formula for flag manifold of type $A$ from wall-crossing
本文考虑 $A$ 型旗流形上的等变积分，利用受 Takuro Mochizuki 穿墙公式启发的计算方法，重新证明了 Weber 和 Zielenkiewicz 的留数公式，并通过设置 $K$ 理论类导出了 Grothendieck 多项式的行列式公式。
In this article, the comodule structure of Chow rings of Flag manifolds $\operatorname{CH}(G/B)$ is described by Schubert cells. Its equivariant version gives rise to a Hopf structure of the equivariant cohomology of flag manifolds $H^*_B(G/B)$. We get two identities of generalized Schubert polynomials as explanations of the geometric facts.
We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivariant bundles over complex homogeneous spaces. Furthermore, we find several explicit formulae for the global height of any generalized flag variety.
We prove a type-uniform Chevalley formula for multiplication with divisor classes in the equivariant quantum $K$-theory ring of any cominuscule flag variety $G/P$. We also prove that multiplication with divisor classes determines the equivariant quantum $K$-theory of arbitrary flag varieties. These results prove a conjecture of Gorbounov and Korff concerning the equivariant quantum $K$-theory of Grassmannians of Lie type A.
We introduce and study various categories of (equivariant) motives of (versal) flag varieties. We relate these categories with certain categories of parabolic (Demazure) modules. We show that the motivic decomposition type of a versal flag variety depends on the direct sum decomposition type of the parabolic module. To do this we use localization techniques of Kostant-Kumar in the context of generalized oriented cohomology as well as the Rost nilpotence principle for algebraic cobordism and its generic version. As an application, we obtain new proofs and examples of indecomposable Chow motives…
Positivity in Weighted Flag Varieties
本文从李理论视角研究加权旗流形的环面等变上同调，证明其正性性质并推广了加权格拉斯曼流形及 A 型情形下的 Chevalley 公式、不动点限制与 GKM 描述等结果。
Positivity in $T$-Equivariant $K$-theory of partial flag varieties associated to Kac-Moody groups
本文证明了与任意 Kac-Moody 群相关的部分旗流形 $G/P$ 的环面等变 $K$-理论中乘积结构常数的符号交替性，将 Kumar 关于 $G/B$ 的先前工作推广至 $G/P$。
We consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau-Ginzburg mirror for that partial flag variety. In our construction, the solutions are labeled by elements of the $K$-theory algebra of the partial flag variety. To establish these facts we consider the equivariant quantum differential equations for a partial flag variety and introduce a compatible system of difference equations, which we call the qKZ equations. We construct a basis of solu…
For a semi-simple simply connected algebraic group G we introduce certain parabolic analogues of the Nekrasov partition function (introduced by Nekrasov and studied recently by Nekrasov-Okounkov and Nakajima-Yoshioka for G=SL(n)). These functions count (roughly speaking) principal G-bundles on the projective plane with a trivialization at infinity and with a parabolic structure at the horizontal line. When the above parabolic subgroup is a Borel subgroup we show that the corresponding partition function is basically equal to the Whittaker matrix coefficient in the universal Verma module over c…
A dual basis for the equivariant quantum $K$-theory of cominuscule varieties
本文研究次微小旗簇等变量子 $K$-理论，定义与 Schubert 基关于量子 $K$-度量对偶的量子化理想层基，给出类型统一组合公式并用于计算 Grassmann 簇的 Schubert 结构常数。
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.
A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part II: quantum double Grothendieck polynomials
本文证明了量子双重Grothendieck多项式在A型旗流形的环面等变量子K-理论环的呈现中代表对偶Schubert类，其关键工具是半无限Schubert类的显式公式。
Given an equivariant oriented cohomology theory $h$, a split reductive group $G$, a maximal torus $T$ in $G$, and a parabolic subgroup $P$ containing $T$, we explain how the $T$-equivariant oriented cohomology ring $h_T(G/P)$ can be identified with the dual of a coalgebra defined using exclusively the root datum of $(G,T)$, a set of simple roots defining $P$ and the formal group law of $h$. In two papers [Push-pull operators on the formal affine Demazure algebra and its dual, arXiv:1312.0019] and [A coproduct structure on the formal affine Demazure algebra, arXiv:1209.1676], we studied the pro…
Using a transversality argument, we demonstrate the positivity of certain coefficients in the equivariant cohomology and K-theory of a generalized flag manifold. This strengthens earlier equivariant positivity theorems (of Graham and Anderson-Griffeth-Miller) by further constraining the roots which can appear in these coefficients. As an application, we deduce that structure constants for comultiplication in the equivariant K-theory of an infinite flag manifold exhibit an unusual positivity property, establishing conjectures of Lam-Lee-Shimozono. Along the way, we present alternative formulas …
We discuss a relationship between Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds, Fomin-Kirillov algebra, and the generalized nil-Hecke algebra. We show that nonnegativity conjecture in Fomin-Kirillov algebra implies the nonnegativity of the Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of the Bott-Samelson varieties. W…
We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$, where $G = GL(n,\C)$, and where $K$ is one of the symmetric subgroups $O(n,\C)$ or $Sp(n,\C)$. We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formu…
Integral equivariant cohomology of affine Grassmannians
本文给出A型仿射格拉斯曼流形和旗流形的整等变上同调的显式表示，并将其与Lam和Shimozono关于有理等变上同调及Larson关于$\nmathbb{P}^1$向量丛模空间整上同调的结果进行比较。
The quantum double Schubert polynomials studied by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, are shown to represent Schubert classes in Kim's presentation of the equivariant quantum cohomology of the flag variety. We define parabolic analogues of quantum double Schubert polynomials, and show that they represent Schubert classes in the equivariant quantum cohomology of partial flag varieties. For complete flags Anderson and Chen have announced a proof with different methods.
Equivariant rigidity of Richardson varieties
证明旗流形中 Schubert 簇和 Richardson 簇由等变上同调类唯一确定，用此结果验证含 Seidel 类的量子上同调乘积对应的两点曲线邻域是 Schubert 簇，提出蕴含等变量子 K-理论 Seidel 乘法公式的猜想并对余迷向旗流形证明该猜想。
Infinite flags and Schubert polynomials
本文借助无限维旗簇与退化轨迹几何研究 Schubert 多项式，证明相关等变余积系数的 Graham 正性并给出多项式展开，同时构造 C 型旗簇嵌入及其上同调拉回映射。
We prove the Givental conjecture that the (equivariant) quantum cohomology of flag manifolds G/B are governed by the conservation law of Toda lattices. In addition, we find the quantum differential module structure of the flag manifolds.
Quantum integrable model for the quantum cohomology/K-theory of flag varieties and the double $β$-Grothendieck polynomials
本文提出推广非对称五顶点自旋链的GL$(n)$量子可积系统，证明其可编码旗簇等变量子上同调与K-理论的环关系，且系统贝特假设态生成双$β$-格罗滕迪克多项式。
We prove a conjecture of Dale Peterson on positivity in the multiplication in the T-equivariant cohomology of the flag variety. The theorem follows from a more general positivity result about the equivariant cohomology of varieties with actions of a solvable group with finitely many orbits. This more general result is an equivariant version of a theorem of Kumar and Nori.
We interpret the GL_n equivariant cohomology of a partial flag variety of flags of length N in \C^n as the Bethe algebra of a suitable gl_N[t] module associated with the tensor power (\C^N)^{\otimes n}.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。