Weyl group symmetries of the toric variety associated with Weyl chambers

For any crystallographic root system, let $W$ be the associated Weyl group, and let $WP$ be the weight polytope (also known as the $W$-permutohedron) associated with an arbitrary strongly dominant weight. The action of $W$ on $WP$ induces an action on the toric variety $X(WP)$ associated with the normal fan of $WP$, and hence an action on the rational cohomology ring $H^{\ast }(X(WP))$. Let $P$ be the corresponding dominant weight polytope, which is a fundamental region of the $W$-action on $WP$. We give a type uniform algebraic proof that the fixed subring $H^{\ast }{(X(WP))}^W$ is isomorphic to the cohomology ring $H^{\ast }(X(P))$ of the toric variety $X(P)$ associated with the normal fan of $P$. Notably, our proof applies to all finite (not necessarily crystallographic) Coxeter groups, answering a question of Horiguchi–Masuda–Shareshian–Song about non-crystallographic root systems.