Poincare polynomials of generic torus orbit closures in Schubert varieties

Generic torus orbit closures in Schubert varieties

The closure of a generic torus orbit in the flag variety $G/B$ of type $A_{n-1}$ is known to be a permutohedral variety and well studied. In this paper we introduce the notion of a generic torus orbit in the Schubert variety $X_w$ $(w\in \mathfrak{S}_n)$ and study its closure $Y_w$. We identify the maximal cone in the fan of $Y_w$ corresponding to a fixed point $uB$ $(u\le w)$, associate a graph $Γ_w(u)$ to each $u\le w$, and show that $Y_w$ is smooth at $uB$ if and only if $Γ_w(u)$ is a forest. We also introduce a polynomial $A_w(t)$ for each $w$, which agrees with the Eulerian polynomial whe…

格拉斯曼型舒伯特簇的拟对称模拟

A quasisymmetric analog of Grassmannian Schubert varieties

本文证明格拉斯曼流形中环面理查森簇的上同调环为拟对称函数环的有限截断，构造仿射铺砌，并利用直接极限得到具有弱H-群结构的ind-簇，其上同调为拟对称函数的Hopf代数，同时推出碎片多面体的f-向量对数凹性。

Affine pavings of regular nilpotent Hessenberg varieties and intersection theory of the Peterson variety

This paper describes a paving by affines for regular nilpotent Hessenberg varieties in all Lie types, namely a kind of cell decomposition that can be used to compute homology despite its weak closure conditions. Precup recently proved a stronger result; we include ours because we use different methods. We then use this paving to prove that the homology of the Peterson variety injects into the homology of the full flag variety. The proof uses intersection theory and expands the class of the Peterson variety in the homology of the flag variety in terms of the basis of Schubert classes. We explic…

Schubert decompositions for ind-varieties of generalized flags

Let $\mathbf{G}$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$ and $\mathbf{P}\subset \mathbf{G}$ be a splitting parabolic ind-subgroup. The ind-variety $\mathbf{G}/\mathbf{P}$ has been identified with an ind-variety of generalized flags in the paper "Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups" (Int. Math. Res. Not. 2004, no. 55, 2935--2953) by I. Dimitrov and I. Penkov. In the present paper we define a Schubert cell on $\mathbf{G}/\mathbf{P}$ as a $\mathbf{B}$-orbit on $\mathbf{G}/\mathbf{P}$, where $\mathbf{B}$ is any Borel ind-subgro…

Peterson Schubert 演算的正性

Positivity of Peterson Schubert Calculus

本文研究 Peterson 簇的等变上同调，证明由任意 Coxeter 元索引的 Schubert 类等变拉回在除以相交重数后构成一个 Z-基，并证明其结构常数具有 Graham 正性及嵌入稳定性，从而在任意 Lie 型下几何地解释了 A 型中的正性结果。