Toric orbifolds associated with partitioned weight polytopes in classical types

Given a root system Φ of type A n , B n , C n , or D n in Euclidean space E , let W be the associated Weyl group. For a point p ∈ E not orthogonal to any of the roots in Φ , we consider the W -permutohedron P W , which is the convex hull of the W -orbit of p . The representation of W on the rational cohomology ring H ∗ ( X Φ ) of the toric variety X Φ associated to (the normal fan to) P W has been studied by various authors. Let { s 1 , … , s n } be a complete set of simple reflections in W . For K ⊆ [ n ] , let W K be the standard parabolic subgroup of W generated by { s k : k ∈ K } . We show that the fixed subring H ∗ ( X Φ ) W K is isomorphic to the cohomology ring of the toric variety X Φ ( K ) associated to a polytope obtained by intersecting P W with half-spaces bounded by reflecting hyperplanes for the given generators of W K . We also obtain explicit formulas for h -vectors of these polytopes. By a result of Balibanu–Crooks, the cohomology rings H ∗ ( X Φ ( K ) ) are isomorphic with cohomology rings of certain regular Hessenberg varieties.