Resumen de Toric orbifolds associated with partitioned weight polytopes in classical types

Tatsuya Horiguchi, Mikiya Masuda, John Shareshian, Jongbaek Song

• Given a root system Φ of type An, Bn, Cn, or Dn 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 PW , which is the convex hull of the Worbit of p. The representation of W on the rational cohomology ring H∗(XΦ) of the toric variety X Φ associated to (the normal fan to) PW has been studied by various authors. Let {s1,...,sn} be a complete set of simple reflections in W. For K ⊆ [n], let WK be the standard parabolic subgroup of W generated by {sk : k ∈ K}. We show that the fixed subring H∗(XΦ)WK is isomorphic to the cohomology ring of the toric variety XΦ (K) associated to a polytope obtained by intersecting PW with half-spaces bounded by reflecting hyperplanes for the given generators of WK . 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.