Generic torus orbit closures in Schubert varieties

TL;DR

This paper introduces a new class of orbit closures within Schubert varieties, characterizes their smoothness via associated graphs, and relates their topological invariants to Eulerian polynomials, expanding understanding of torus actions in algebraic geometry.

Contribution

It defines generic torus orbit closures in Schubert varieties, characterizes their smoothness through graph-theoretic conditions, and links their Poincaré polynomials to Eulerian polynomials.

Findings

Y_w is smooth at uB iff Γ_w(u) is a forest.

The Poincaré polynomial of Y_w matches A_w(t^2) when Y_w is smooth.

A_w(t) generalizes the Eulerian polynomial for all w.

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 Poincar\'e polynomial of $Y_w$ agrees with $A_w(t^2)$ when $Y_w$ is smooth.