A Chevalley formula in equivariant $K$-theory
Matthieu Willems
本文给出了旗簇的等变$K$ -理论中Schubert簇的结构层与线丛相乘的递推公式。
The aim of this paper is to give a recursive formula to multiply a line bundle with the structure sheaf of a schubert variety in the equivariant $K$-theory of a flag variety.
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。
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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.
Toroidal embedding of Chevalley groups over $\mathbb{Z}$
本文证明了$\mathbb{Z}$上分裂约化群概形存在“万有”等变环面嵌入，使其通过基变换特殊化为经典环面嵌入，并讨论了其几何性质。
Equivariant K-Chevalley Rules for Kac-Moody Flag Manifolds
在Kac-Moody旗流形的环面等变K-理论中，给出了等变线丛类与Schubert类乘积的组合无消规则.线丛的权可以是显性的，也可以是反显性的，其系数可以用Lakshmibai-Seshadri路径描述，也可以用第一作者和Postnikov的凹室模型描述.对于Lakshmibai-Seshadri路径，我们的公式是Griffeth-Ram和Pittie-Ram关于有限维旗流形结果的Kac-Moody推广.所提到的结果的证明中的差距得到解决。
Given a finite group of automorphisms of a compact Riemann surface, the Chevalley-Weil formula computes the character valued Euler characteristic of an equivariant line bundle. The goal of this article is to give a proof by computing using residues of a Gauss-Manin connection.
Equivariant Chevalley, Giambelli, and Monk Formulae for the Peterson Variety
本文提出了一种基于 Chern 类的多项式公式，用于计算 Peterson 变形与 Schubert 变形的交集乘数，并发展了等变 Chevalley 公式和 dual Monk 规则。
A fixed point formula of Lefschetz type in Arakelov geometry III: representations of Chevalley schemes and heights of flag varieties
给出了Spec Z上Chevalley格式的整表示的Jantzen和公式的一个新的证明.这是通过将Arakelov几何中Lefschetz型的不动点公式应用于广义旗簇来实现的。我们的证明涉及到复齐性空间上所有等变丛的等变Ray-Singer挠率的计算。此外，我们找到了几个显式公式的任何广义旗簇的整体高度。
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…
Computing the equivariant Euler characteristic of Zariski and etale sheaves on curves
证明了一个等变的Grothendieck-Ogg-Shafarevich公式。这个公式可以被看作是一个著名的公式的经典类似物，推广了经典的Chevalley-Weil公式。我们给出了一个新的方法，这些公式（首先证明了Ellingsrud/Lønsted，Nakajima，Kani和Ksir），也可以应用在étale的情况下。
A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory
本文利用量子壁龛模型建立半无限旗流形等变K-群的任意权重Chevalley公式，并由此证明普通旗流形量子K-理论中反支配基本权重的Chevalley公式，解决长期猜想，同时在A型中刻画量子Grothendieck多项式与Schubert类的关系。
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.
The Chevalley-Weil formula on nodal curves
本文研究了代数闭域上节点曲线全纯微分空间在有限自同构群作用下的特征子空间，在商曲线附加条件下计算了 Chevalley-Weil 公式并给出实例。
We continue the study, begun in [Kouno-Naito-Orr-Sagaki, 2021], of inverse Chevalley formulas for the equivariant $K$-group of semi-infinite flag manifolds. Using the language of alcove paths, we reformulate and extend our combinatorial inverse Chevalley formula to arbitrary weights in all simply-laced types (conjecturally also for $E_8$).
We give a Pieri rule for the torus-equivariant cohomology of (submaximal) Grassmannians of Lie types B, C, and D. To the authors' best knowledge, our rule is the first manifestly positive formula, beyond the equivariant Chevalley formula. We also give a simple proof of the equivariant Pieri rule for the ordinary (type A) Grassmannian.
In our previous paper, we studied the category of semifinite bundles on a proper variety defined over a field of characteristic 0. As a result, we obtained the fact that for a smooth projective curve defined over an algebraically closed field of characteristic 0 with genus $g>1$, Nori fundamental group acts faithfully on the unipotent fundamental group of its universal covering. However, it was not mentioned about any explicit module structure. In this paper, we prove that the Chevalley-Weil formula gives a description of it.
Equivariant quantum cohomology of homogeneous spaces
本文证明了齐次簇X=G/P中两个Schubert类的等变量子乘法的Chevalley公式。当X是Grassmannian时，这个公式包含了计算X的等变量子上同调代数的结构常数的有效算法。
Chevalley-Monk formulas for bow varieties
本文证明了A型bow variety的tautological bundle的等变第一Chern类在stable envelope基下的乘法公式，推广了经典Chevalley-Monk公式，并可用标记stable envelope基的skein型图的交叉来表述。
Quantum K-theory Chevalley formulas in the parabolic case
本文利用任意 G/B 旗流形 T-等变量子 K-理论中基于量子凹室模型的统一 Chevalley 公式，推导出 A、C 型 Grassmann 流形及 A 型两步旗流形的无消去 Chevalley 型乘法公式。
A positive Monk formula in the S^1-equivariant cohomology of type A Peterson varieties
彼得森簇是海森伯格簇的一个特殊类别，已经被广泛研究，例如彼得森，Kostant和Rietsch，与旗簇的量子上同调有关。在这篇手稿中，我们开发了一个广义舒伯特演算，特别是一个积极的Chevalley-Monk公式，为普通和Borel等变上同调的彼得森品种$Y$类型$A_{n-1}$，关于自然的$S^1$ -作用所产生的标准作用的极大环面旗品种。据我们所知，这是第一个超越Kac-Moody旗形变量$G/P$的正舒伯特演算的例子。我们的主要结果如下。首先，我们确定一个计算方便的基础$H^*_{S^1}(Y)$，我们称之为彼得森舒伯特类的基础。其次，我们得到了一个显着的积极的，积分Chevalley-Monk公式的产品的上同调度2彼得森舒伯特类与任意的彼得森舒伯特类。$H^*_{S^1}(Y)$和$H^*(Y)$均以2度生成。最后，利用我们的Chevalley-Monk公式给出了$A_{n-1}$型Peterson簇的$S^1$ -等变上同调环$H^*_{S^1}(Y)$和普通上同调环$H^*(Y)$的显式描述（通过生成元和关系）.我们的方法都是直接从GKM（Goresky-Kottwitz-MacPherson）理论和经典的舒伯特微积分和启发。我们讨论了几个开放的问题和未来的工作方向。
In the 1930s Chevalley and Weil gave a formula for decomposing the canonical representation on the space of differential forms of the Galois group of a ramified Galois cover of Riemann surfaces. In this article we prove an analogous Chevalley-Weil formula for ramified Galois covers of orbifold curves. We then specialize the formula to the case when the base orbifold curve is the (reduced) modular orbifold. As an application of this latter formula we decompose the canonical representations of modular curves of full, prime level and of Fermat curves of arbitrary exponent.
Chevalley's ambiguous class number formula for an arbitrary torus
这个版本是对原始论文的重大改进。它包括一个新的部分，我们讨论规范环面的一些细节。新的摘要如下：本文得到了整体域上任意环面T的Chevalley的模糊类数公式。在我们的公式中设T= G_{m}，可以恢复Chevalley的经典公式.作为一般结果的一个例子，我们详细讨论了范数环面。一个关键成分的证明我们的主要定理是工作的X.Xarles组的组成部分的奈隆雷诺模型的环面。
We study the moduli stack of degree $0$ semistable $G$-bundles on an irreducible curve $E$ of arithmetic genus $1$, where $G$ is a connected reductive group. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups $H$ of $G$ (the $E$-pseudo-Levi subgroups), where each stratum is computed in terms of $H$-bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan--Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where $E$ has a single cusp (res…
The bisymplectic Grassmannian I$_2$Gr$(k, V)$ parametrizes k-dimensional subspaces of a vector space V which are isotropic with respect to two general skew-symmetric forms; it is a Fano variety which admits an action of a torus with a finite number of fixed points. In this work we study its equivariant cohomology when $k = 2$; the central result of the paper is an equivariant Chevalley formula for the multiplication of the hyper-plane class by any Schubert class. Moreover, we study in detail the case of I$_2$Gr$(2, \mathbb{C}^6)$, which is a quasi-homogeneous variety, we analyze its deformatio…
Restriction theorems: from orbits and Chevalley to periods and Galois
利用Galois理论的新方法，研究约化群复表示中满足不变环和函数域限制性质的子簇，并应用于Calabi-Yau族的周期积分显式公式。
Peterson-Lam-Shimozono's theorem is an affine analogue of quantum Chevalley formula
本文为Peterson-Lam-Shimozono定理提供新证明，通过构造并识别Gromov-Witten理论中的代数同态为Peterson映射，完全确定了联系旗簇等变量子同调与仿射Grassmannian等变Pontryagin同调的结构常数。
The odd symplectic Grassmannian $\mathrm{IG}:=\mathrm{IG}(k, 2n+1)$ parametrizes $k$ dimensional subspaces of $\mathbb{C}^{2n+1}$ which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on $\mathrm{IG}$ with two orbits, and $\mathrm{IG}$ is itself a smooth Schubert variety in the submaximal isotropic Grassmannian $\mathrm{IG}(k, 2n+2)$. We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of $\mathrm{IG}$, i.e. a formula to multiply a Schubert class by the Schubert divisor cl…
On the cohomological representations of finite automorphism groups of singular curves and compact complex spaces
本文研究了有限群作用下奇异曲线和紧复空间的上同调表示，推导了Chevalley--Weil型公式，并探讨了节点曲线的不变截面和变形空间。
Kazhdan-Laumon representations of finite Chevalley groups, character sheaves and some generalization of the Lefschetz-Verdeier trace formula
本文研究了有限域上约化群的表示。我们表明，一般这些表示是不可约的，他们的字符是等于由弗罗贝尼乌斯态射的相应的Lusztig的特征层的茎的痕迹得到的功能。这与G.Lusztig的相应结果一起意味着一般的Kazhdan-Laumon表示同构于相应的Deligne-Lusztig表示。在本文的后半部分，我们给出了一个直接的尺规作图法，这个同构，假设某些推广的Lefschetz-Verdier迹公式持有。
The Chevalley--Weil formula for finite group actions on higher dimensional compact complex manifolds
本文基于 Atiyah--Singer 全纯 Lefschetz 不动点定理定义了分歧模，将紧 Riemann 面的 Chevalley--Weil 公式推广至高维紧复流形上的有限群作用，并给出了相关公式与计算实例。
Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes
本文基于Lenart和Postnikov的λ链，推导了广义旗 manifold 中 Schubert 胞 motivic Chern 类与线丛类相乘的 Chevalley 公式，并探讨了稳定包在 T*(G/B) 中的墙穿越公式及多项式系数的正性性质。
Affine Weyl groups in K-theory and representation theory
给出了广义旗簇G/P的等变K-理论的一个显式组合Chevalley型公式，它是经典Chevalley公式的直接推广.我们的公式意味着一个简单的组合模型的字符的不可约表示的G，更一般地说，Demazure字符。该模型可以被看作是利特尔曼路径模型的离散对应物，并且具有几个优点。我们的构造是根据一定的R-矩阵，即满足杨-巴克斯特方程的算子集合给出的。它减少到组合学的分解在仿射外尔群和计数饱和链的Bruhat秩序的（非仿射）外尔群。我们的模型很容易暗示几个对称的系数在Chevalley型公式。我们还推导出一个简单的公式乘以任意舒伯特类的除数类，以及对偶Chevalley型公式。该文件包含其他应用程序和例子。
Equivariant Giambelli and determinantal restriction formulas for the Grassmannian
本文的主要结果是Grassmannian环面等变积分上同调环中Schubert子簇的等变类对环面不动点的限制的一个行列式公式。作为推论，我们得到了Giambelli公式的一个等变形式。
We give a summation over graphs type formula for the permutation-equivariant K-theoretic Gromov-Witten total potential of a projective manifold X in terms of cohomological Gromov-Witten (GW) invariants of X. We achieve this by describing combinatorially the Kawasaki strata of the moduli spaces of stable maps to X.
We prove a formula of the equivariant infinity-adic special L-values of abelian t-modules. This gives function field analogues of the equivariant class number formula. As an application, we calculate the special values of Artin L-functions for Galois representations.
On the S_n-equivariant Euler characteristic of M_{2,n}
Getzler公式将簇X上不同点的有序元组空间的S_n-等变Hodge-Deligne多项式与X的Hodge-Deligne多项式联系起来。我们得到类似的情况下，当X有一个非平凡的自同构群这个公式。将具有不同自同构群的$\mathcal{M}_2$的所有层集合在一起，导出了$\mathcal{M}_{2,n}$的S_n-等变欧拉示性数的计算公式.
On a formula for the equivariant Euler characteristic of a $G$-sheaf
本文利用商栈的Riemann-Roch定理，将域上$G$-曲线$X$上$G$-层等变Euler示性数的计算推广到了$dim(X)>1$的高维情形。
Billey and Braden defined a geometric pattern map on flag manifolds which extends the generalized pattern map of Billey and Postnikov on Weyl groups. The interaction of this torus equivariant map with the Bruhat order and its action on line bundles lead to formulas for its pullback on the equivariant cohomology ring and on equivariant K-theory. These formulas are in terms of the Borel presentation, the basis of Schubert classes, and localization at torus fixed points.
We continue our study of the genus-$0$ permutation-equivariant quantum K-theory of the target $X=pt$, and completely determine the "big J-function" of this theory. The computation is based on the application of Lefschetz' fixed point formula to the action of $S_n$ on $\overline{M}_{0,n+1}$. It is an instance of the general "adelic characterization" (which we state at the end with reference to arXiv:1106.3136) of quantum K-theory for any target $X$ in terms of quantum cohomology theory. Yet, some simplifications of non-conceptual nature occur in this example, making it a lucid illustration to t…
Chevalley-Weil formula for hypersurfaces in $\mathbf{P}^n$-bundles over curves and Mordell-Weil ranks in function field towers
本文研究曲线上射影丛中超曲面在 Galois 基变换下的 Hodge 上同调群结构，给出 Chevalley-Weil 型公式，并借此几何证明椭圆曲面 Mordell-Weil 秩的 Pal 上界，同时比较其与 Shioda-Tate 公式界的强弱关系。
Equivariant Todd Classes for Toric Varieties
对于完全复曲面簇，我们得到了局部化等变Todd类在组合数据--扇形下的一个显式公式。这是基于等变Riemann-Roch定理和环面簇的等变上同调和等变同调的计算。
On the quantum product of Schubert classes
我们给出了一个公式的最小权力的量子参数q发生在产品的舒伯特类的（小）量子上同调的一般标志品种G/P.我们还包括一个完整的证明彼得森的量子版本Chevalley的公式，也为一般G/P的。
For a $G$-scheme $X$ with a given equivariant perfect obstruction theory, we prove a virtual equivariant Grothendieck-Riemann-Roch formula, this is an extension of a result of Fantechi-Göttsche to the equivariant context. We also prove a virtual non-abelian localization theorem for schemes over $\mathbb{C}$ with proper actions.
A Pieri-type Formula for the Equivariant Cohomology of the Flag Manifold
我们证明了旗流形SLn/B的T-等变上同调的某些结构常数的一个显式组合公式。我们的结果推广了Sottile在1996年证明的关于普通上同调的Pieri型公式。我们的结果也给出了由Lascoux和Schutzenberger引入的双Schubert多项式的Pieri型公式。
Curve neighborhoods of Schubert varieties
以前的结果，作者与Chaput和佩兰指出，工会的所有有理曲线的固定度通过舒伯特品种在一个齐性空间G/P再次舒伯特品种。本文利用外尔群元素的Hecke积明确地确定了这个Schubert簇。应用我们的结果给出了任意两点Gromov-Witten不变量的一个显式公式，以及量子Chevalley公式及其等变推广的一个新证明.我们还恢复了一个公式的最小程度的有理曲线之间的两个给定点的一个cominuscule品种。
We use Young's raising operators to introduce and study double eta polynomials, which are an even orthogonal analogue of Wilson's double theta polynomials. Our double eta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of even orthogonal Grassmannians, and specialize to the single eta polynomials of Buch, Kresch, and the author.
The aim of this paper is to describe how to obtain residue-type formulas for push-forwards in equivariant cohomology, using the Jeffrey-Kirwan nonabelian localization theorem and the related result of Guillemin and Kalkman. This paper relates the description of equivariant push-forwards for homogeneous spaces of classical Lie groups with the action of the maximal torus with symplectic geometry, using the example of projective spaces and classical Grassmannians.
In a previous paper Affine symmetries of the equivariant quantum cohomology of rational homogeneous spaces, a general formula was given for the multiplication by some special Schubert classes in the quantum cohomology of any homogeneous space. Although this formula is correct in the non equivariant setting, the stated equivariant version was wrong. We provide corrections for the equivariant formula, thus giving a correct argument for the non equivariant formula. We also give new formulas in the equivariant homology of the affine grassmannian that could lead to Pieri type formulas.
Schubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Lagrangian Grassmannian
给出了Schubert类在等变K-理论和Lagrange Grassmannian的等变上同调中对T-不动点的限制的正公式。我们的公式依赖于Ghorpade-Raghavan的一个结果，该结果给出了Schubert簇在Lagrangian Grassmannian的T-不动点附近的等变Grobner退化。
We use Young's raising operators to introduce and study double theta polynomials, which specialize to both the theta polynomials of Buch, Kresch, and Tamvakis, and to double (or factorial) Schur S-polynomials and Q-polynomials. These double theta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of symplectic Grassmannians, and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations.
Excited Young diagrams and equivariant Schubert calculus
利用到环面不动点的局部化映射刻画了各向同性Grassmannian的环面等变上同调环.给出了这类齐性空间的等变Schubert类的两类公式。第一个公式涉及的组合对象，我们称之为“兴奋的杨氏”和第二个是书面的阶乘舒尔$Q$ -或$P$ -功能。作为应用，我们给出了等变Schubert类的Giambelli型公式。我们还给出了Schubert簇中奇点重数的组合公式和Pfiran公式。
Schubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Grassmannian
在等变K-理论和Grassmannian的等变上同调中，我们给出了Schubert类对T-不动点的限制的正公式.我们的公式依赖于Kodiyalam-Raghavan和Kreiman-Lakshmibai的一个结果，该结果给出了Schubert簇在Grassmannian的T-不动点邻域内的等变Grobner退化.
We give a new proof that three families of polynomials coincide: the double Schubert polynomials of Lascoux and Schützenberger defined by divided difference operators, the pipe dream polynomials of Bergeron and Billey, and the equivariant cohomology classes of matrix Schubert varieties. All three families are shown to satisfy a "co-transition formula" which we explain to be some extent projectively dual to Lascoux' transition formula. We comment on the K-theoretic extensions.
Equivariant Jeffrey-Kirwan localization theorem in non-compact setting
我们将Jeffrey-Kirwan局部化定理推广到非紧的辛和超Kahler代数。类似于Hausel和Proudfoot的圆紧积分，我们使用Atiyah-Bott-Berline-Vergne局部化公式作为形式定义来定义非紧流形上的等变积分。我们引入了一个所谓的等变Jeffrey-Kirwan剩余，我们表明，它具有类似的性质，通常的。我们的局部化公式与通常的Jeffrey-Kirwan公式具有相同的结构，但它使用了形式积分和等变剩余。我们也给了一个版本的hyperKahler的。最后，我们应用这个公式计算了平面上点的Hilbert概型的等变上同调环。
K-theoretic Heisenberg algebras and permutation-equivariant Gromov--Witten theory
将K-理论Heisenberg代数应用于置换等变Gromov--Witten理论，并给出亏格0关联子的显式公式。
Plücker formulas using equivariant cohomology of coincident root strata
本文提出利用等变上同调计算重合根轨迹的万有上同调类，证明广义Plücker公式为次数的多项式，并给出其次数上界及达到上界时的首项，展示了等变方法在枚举几何中的应用。
Nonabelian localization in equivariant K-theory and Riemann-Roch for quotients
证明了光滑代数空间上任意复代数群作用于有限稳定子的等变代数$K$ -理论中的一个局部化公式。这将局部化公式H. A. Nielsen在等变$K$ -向量丛理论和R.W.高阶$K$的基础-等变相干层理论。作为应用，我们给出了光滑代数空间中由真群作用生成的Riemann-Roch公式.这个公式推广了B. Toen关于具有拟投射模空间的栈和作者关于可对角化群的等价物。
A Giambelli formula for the $S^1$-equivariant cohomology of type A Peterson varieties
本文的主要结果是$A$型Peterson簇的$S^1$ -等变上同调环中Peterson Schubert类的Giambelli公式。我们的结果依赖于蒙克公式的等变结构常数的彼得森舒伯特类原田和Tymoczko。此外，我们还证明了H.成濑：第一，Peterson簇的$S^1$ -等变上同调的乘法结构中出现的某些常数是第二类Stirling数;第二，Peterson Schubert类在旗簇的$T$ -等变上同调中满足与经典等变Schubert类的稳定性类似的稳定性性质.
We find an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL(2) Higgs moduli space on a Riemann surface. On one side we have the components of the Lagrangian brane of U(1,1) Higgs bundles whose mirror was proposed by Nigel Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL(2) Higgs moduli space. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin's proposal.
Localization in equivariant intersection theory and the Bott residue formula
本文证明了等变交理论中环面作用的局部化定理。利用这个定理，我们给出了光滑完备簇上丛的陈数的Bott留数公式的另一个证明。此外，我们的技术使我们能够获得的剩余公式束上的某一类奇异计划，承认环面行动。这个类是相当特殊的，但它包括一些有趣的例子，如完全交叉和舒伯特品种。
For any Bott-Samelson resolution $q_{I}:\hat{X_{I}}\rightarrow G/B$ of the flag variety $G/B$, and any torus equivariant oriented cohomology $h_T$, we compute the restriction formula of certain basis $η_L$ of $h_T(\hat{X_{I}})$ determined by the projective bundle formula. As an application, we show that $h_T(\hat{X_{I}})$ embeds into the equivariant oriented cohomology of $T$-fixed points, and the image can be characterized by using the Goresky-Kottwitz-MacPherson (GKM) description. Furthermore, we compute the push-forward of the basis $η_L$ onto $h_T(G/B)$, and their restriction formula.
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.
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