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