Generic torus orbit closures in Schubert varieties

Abstract

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 ${\Gamma }_w(u)$ to each $u\le w$, and show that $Y_w$ is smooth at $uB$ if and only if ${\Gamma }_w(u)$ is a forest. We also introduce a polynomial $A_w(t)$ for each $w$, which agrees with the Eulerian polynomial when $w$ is the longest element of ${\mathfrak{S}}_n$, and show that the Poincaré polynomial of $Y_w$ agrees with $A_w(t^2)$ when $Y_w$ is smooth.

References in corpus (4)

• Semifree circle actions, Bott towers, and quasitoric manifolds
• On the Integral Cohomology Ring of Toric Orbifolds and Singular Toric Varieties
• Y-systems and generalized associahedra
• Toric Bruhat interval polytopes

Cited by in corpus (9)

• Toric Bruhat interval polytopes
• Flag Bott manifolds and the toric closure of a generic orbit associated to a generalized Bott manifold
• GKM theory for orbifold stratified spaces and application to singular toric varieties
• On Schubert varieties of complexity one
• The smooth torus orbit closures in the Grassmannians
• Gorenstein braid cones and crepant resolutions
• One-skeleton posets of Bruhat interval polytopes
• Generic torus orbit closures in flag Bott manifolds
• Poincare polynomials of generic torus orbit closures in Schubert varieties