# Toric Bruhat interval polytopes

## TL;DR

This paper characterizes when Bruhat interval polytopes are toric and when they are cubes, linking these properties to the combinatorial structure of the Bruhat interval and the smoothness of associated toric varieties.

## Contribution

It establishes that toric Bruhat interval polytopes are determined by Bruhat interval posets and characterizes cube-shaped polytopes via Boolean algebra structures and toric smoothness.

## Key findings

- Toric Bruhat interval polytopes are determined by Bruhat interval posets.
- A Bruhat interval polytope is a cube if and only if it is toric and the interval is a Boolean algebra.
- Several sufficient conditions for Bruhat interval polytopes to be cubes are provided.

## Abstract

For two elements $v$ and $w$ of the symmetric group $\mathfrak{S}_n$ with $v\leq w$ in Bruhat order, the Bruhat interval polytope $Q_{v,w}$ is the convex hull of the points $(z(1),\ldots,z(n))\in \mathbb{R}^n$ with $v\leq z\leq w$. It is known that the Bruhat interval polytope $Q_{v,w}$ is the moment map image of the Richardson variety $X^{v^{-1}}_{w^{-1}}$. 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.