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.

added results on real points