Abstract. Given a simple graph G, the graph associahedron P G is a convex polytope whose facets correspond to the connected induced subgraphs of G. Graph associahedra have been studied widely and are found in a broad range of subjects. Recently, S. Choi and H. Park computed the rational Betti numbers of the real toric variety corresponding to a graph associahedron under the canonical Delzant realization. In this paper, we focus on a pseudograph associahedron which was introduced by Carr, Devadoss and Forcey, and then discuss how to compute the Poincaré polynomial of the real toric variety corresponding to a pseudograph associahedron under the canonical Delzant realization.
A quasitoric manifold is a 2n-dimensional compact smooth manifold with a locally standard action of an n-dimensional torus whose orbit space is a simple polytope. We classify quasitoric manifolds with second Betti number β 2 = 2 topologically. Interestingly, they are distinguished by their cohomology rings up to homeomorphism.
Regular semisimple Hessenberg varieties are subvarieties of the flag variety Flag(C n ) arising naturally in the intersection of geometry, representation theory, and combinatorics. Recent results of Abe-Horiguchi-Masuda-Murai-Sato and Abe-DeDieu-Galetto-Harada relate the volume polynomials of regular semisimple Hessenberg varieties to the volume polynomial of the Gelfand-Zetlin polytope GZ(λ) for λ = (λ 1 , λ 2 , . . . , λn). The main results of this manuscript use and generalize tools developed by Anderson-Tymoczko, Kiritchenko-Smirnov-Timorin, and Postnikov, in order to derive an explicit formula for the volume polynomials of regular semisimple Hessenberg varieties in terms of the volumes of certain faces of the Gelfand-Zetlin polytope, and also exhibit a manifestly positive, combinatorial formula for their coefficients with respect to the basis of monomials in the α i := λ i − λ i+1 . In addition, motivated by these considerations, we carefully analyze the special case of the permutohedral variety, which is also known as the toric variety associated to Weyl chambers. In this case, we obtain an explicit decomposition of the permutohedron (the moment map image of the permutohedral variety) into combinatorial (n − 1)-cubes, and also give a geometric interpretation of this decomposition by expressing the cohomology class of the permutohedral variety in Flag(C n ) as a sum of the cohomology classes of a certain set of Richardson varieties.
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