Toric Bruhat interval polytopes

Abstract

For two elements $v$ and $w$ of the symmetric group ${\mathfrak{S}}_n$ with $v\le w$ in Bruhat order, the Bruhat interval polytope $Q_{v,w}$ is the convex hull of the points $(z(1),\dots ,z(n))\in {\mathbb{R}}^n$ with $v\le z\le w$. It is known that the Bruhat interval polytope $Q_{v,w}$ is the moment map image of the Richardson variety $X_{w^{-1}}^{v^{-1}}$. We say that $Q_{v,w}$ is \emph{toric} if the corresponding Richardson variety $X_{w^{-1}}^{v^{-1}}$ is a toric variety. We show that when $Q_{v,w}$ is toric, its combinatorial type is determined by the poset structure of the Bruhat interval $[v,w]$ while this is not true unless $Q_{v,w}$ is toric. We are concerned with the problem of when $Q_{v,w}$ is (combinatorially equivalent to) a cube because $Q_{v,w}$ is a cube if and only if $X_{w^{-1}}^{v^{-1}}$ is a smooth toric variety. We show that a Bruhat interval polytope $Q_{v,w}$ is a cube if and only if $Q_{v,w}$ is toric and the Bruhat interval $[v,w]$ is a Boolean algebra. We also give several sufficient conditions on $v$ and $w$ for $Q_{v,w}$ to be a cube.

30 pages, 10 figures

References in corpus (1)

• Generic torus orbit closures in Schubert varieties

Cited by in corpus (6)

• Generic torus orbit closures in Schubert varieties
• Flag Bott manifolds and the toric closure of a generic orbit associated to a generalized Bott manifold
• On Schubert varieties of complexity one
• Polyhedral and Tropical Geometry of Flag Positroids
• Poincare polynomials of generic torus orbit closures in Schubert varieties
• One-skeleton posets of Bruhat interval polytopes