Weyl group symmetries of the toric variety associated with Weyl chambers

Abstract

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 $H^{\ast }\left(X(WP)\right)$. 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 $H^{\ast }{\left(X(WP)\right)}^W$ is isomorphic to the cohomology ring $H^{\ast }\left(X(P)\right)$ 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.

14 pages, final version

References in corpus (1)

• Toric orbifolds associated with partitioned weight polytopes in classical types