Puzzles and (equivariant) cohomology of Grassmannians

Grassmannians 与等变上同调及迷向作用

Grassmannians and the equivariant cohomology of isotropy actions

本文提出一种较简明的方法计算连通 Lie 群的等变上同调环，并以此恢复实 Grassmannian 与实定向 Grassmannian 在迷向作用下的等变上同调。

An explicit determination of the $K$-theoretic structure constants of the affine Grassmannian associated to $SL_2$

Let $G:=\widehat{SL_2}$ denote the affine Kac-Moody group associated to $SL_2$ and $\bar{\mathcal{X}}$ the associated affine Grassmannian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant Grothendieck group of $\bar{\mathcal{X}}$. In the case of ordinary (non-equivariant) $K$-theory we find an explicit closed form for the structure constants. We also determine an inductive formula for the structure constants in the torus-equivariant cohomology ring, and use this formula to find closed forms for some of the structure constants.

同调型的等效射影空间和格拉斯曼流形的有理稳定性类型

Rational stable homotopy type of equivariant projective spaces and Grassmannians

本文研究了等效复射影空间和格拉斯曼流形在复表示下的有理稳定性类型，通过显式分裂展示了其结构特征。

T-equivariant cohomology of cell complexes and the case of infinite Grassmannians

In 1998, Goresky, Kottwitz, and MacPherson showed that for certain spaces X equipped with a torus action, the T-equivariant cohomology ring of X can be described by combinatorial data obtained from its orbit decomposition. Thus, their theory transforms calculations of the equivariant topology of X to those of the combinatorics of the orbit decomposition. Since then, many authors have studied this interplay between topology and combinatorics. In this paper, we generalize the theorem of Goresky, Kottwitz, and MacPherson to the (possibly infinite-dimensional) setting where X is any equivariant ce…

Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology

We define an elliptic version of the stable envelope of Maulik and Okounkov for the equivariant elliptic cohomology of cotangent bundles of Grassmannians. It is a version of the construction proposed by Aganagic and Okounkov and is based on weight functions and shuffle products. We construct an action of the dynamical elliptic quantum group associated with $\mathfrak{gl}_2$ on the equivariant elliptic cohomology of the union of cotangent bundles of Grassmannians. The generators of the elliptic quantum groups act as difference operators on sections of admissible bundles, a notion introduced in …

复二次型的$C_2$-等变普通上同调 I：反对称情形

The $C_2$-equivariant ordinary cohomology of complex quadrics I: The antisymmetric case

本文计算了光滑反对称复二次型的$C_2$-等变普通上同调，并利用其与等变Grassmann流形的关系，证明了$\mathbb{P}^3$中三次曲面上27条直线结论的等变加细。

Multicurves and equivariant cobordism

Let A be a finite abelian group. We set up an algebraic framework for studying A-equivariant complex-orientable cohomology theories in terms of a suitable kind of equivariant formal groups. We compute the equivariant cohomology of many spaces in these terms, including projective bundles (and associated Gysin maps), Thom spaces, and infinite Grassmannians.

双分次庞加莱多项式与 Rep($C_2$)-复形的等变上同调

Bigraded Poincaré polynomials and the equivariant cohomology of Rep($C_2$)-complexes

本文针对 Rep($C_2$)-等变 Grassmannian 的 Bredon 上同调计算，引入取值于 $\mathbb{Z}[x,y]$ 的 $\mathbb{M}_2$-模统计量以降低计算复杂度，并给出若干新结果。

一维 juggling 变种的等变上同调环

Equivariant cohomology of juggling varieties in rank one

本文通过将一维 juggling 变种视为循环有向图 Grassmannian，构造了其等变上同调的 Knutson-Tao 基，描述了环结构并计算了结构常数，证明其为整数。

Quantum multiplication through equivariant Schubert calculus

In this note, we rederive quantum Pieri's formula and the rim hook algorithm in quantum Schubert calculus by studying multiplication in the equivariant cohomology ring of Grassmannians with respect to equivariant Schubert classes which are characteristic classes. We also extend this idea in studying equivariant quantum Schubert calculus, and obtain the equivariant quantum Giambelli's and Pieri's formulae in terms of characteristic classes, with the former formula shown to be free of quantum deformation.

复二次曲面的$C_2$等变普通上同调III：例外情形

The $C_2$-equivariant ordinary cohomology of complex quadrics III: Exceptional cases

本文为$C_2$等变复二次曲面系列终篇，完成不动点集分量超2的光滑对称二次曲面等变普通上同调计算，推导对应格拉斯曼流形结果，证明了三维射影空间三次曲面27条直线结论的等变细化。

Divided difference operators for partial flag varieties

Divided difference operators are degree-reducing operators on the cohomology of flag varieties that are used to compute algebraic invariants of the ring (for instance, structure constants). We identify divided difference operators on the equivariant cohomology of G/P for arbitrary partial flag varieties of arbitrary Lie type, and show how to use them in the ordinary cohomology of G/P. We provide three applications. The first shows that all Schubert classes of partial flag varieties can be generated from a sequence of divided difference operators on the highest-degree Schubert class. The second…

Generalized Equivariant Cohomology and Stratifications

For $T$ a compact torus and $E_T^*$ a generalized $T$-equivariant cohomology theory, we provide a systematic framework for computing $E_T^*$ in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute $E_T^*(X)$ as an $E_T^*(\text{pt})$-module when $X$ is a direct limit of smooth complex projective $T_{\mathbb{C}}$-varieties with finitely many $T$-fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$, $K_T^*$, and $MU_T^*$. We perform this computation on the affine Grassmannian of a complex semisimple group.

Gysin 公式与等变上同调

Gysin formulas and equivariant cohomology

本文以射影丛为例，简化阐述了作者此前基于等变局部化公式给出的纤维丛 Gysin 映射一般计算方法。

Stiefel流形上的$O(k)$-等变降维

$O(k)$-Equivariant Dimensionality Reduction on Stiefel Manifolds

针对高维Stiefel流形数据，提出主Stiefel坐标（PSC）算法，通过PCA扩展和梯度下降两种嵌入方式实现$O(k)$-等变降维，在无噪声时全局最小化投影误差，非理想数据时提升效果，结果可扩展至Grassmann流形。