Toric varieties modulo reflections

Abstract

Let $W$ be a finite group generated by reflections of a lattice $M$. If a lattice polytope $P \subset M \otimes_{\mathbb Z}\mathbb R$ is preserved by $W$, then we show that the quotient of the projective toric variety $X_P$ by $W$ is isomorphic to the toric variety $X_{P \cap D}$, where $D$ is a fundamental domain for the action of $W$. This answers a question of Horiguchi-Masuda-Shareshian-Song, and recovers results of Blume, of Song, of the second author, and of Gui-Hu-Liu. We also study quotients of real toric varieties, proving that $X_P^{\mathbb R} / W$ is contractible when $P$ is a permutohedron.