Toric varieties associated with partitioned weight polytopes

A weight polytope is a convex polytope obtained by taking the convex hull of the Weyl group orbit of a point in a weight space. For the Lie type A, a weight polytope is called a permutohedron. In this talk, we consider the toric variety X associated with a weight polytope together with a natural Weyl group action on it. In particular, we study the invariant part of the cohomology ring of X for the action of a parabolic subgroup of the given Weyl group, which is indeed isomorphic to the cohomology ring of a toric orbifold associated with a ‘partitioned weight polytope’. This result provides explicit cohomology presentations of a certain class of regular Hessenberg varieties. It is a joint work with T. Horiguchi, M. Masuda and J. Shareshian.