No abstract

The antiprism triangulation provides a natural way to subdivide a simplicial complex ∆, similar to barycentric subdivision, which appeared independently in combinatorial algebraic topology and computer science. It can be defined as the simplicial complex of chains of multi-pointed faces of ∆, from a combinatorial point of view, and by successively applying the antiprism construction, or balanced stellar subdivisions, on the faces of ∆, from a geometric point of view.This paper studies enumerative invariants associated to this triangulation, such as the transformation of the h-vector of ∆ under antiprism triangulation, and algebraic properties of its Stanley-Reisner ring. Among other results, it is shown that the h-polynomial of the antiprism triangulation of a simplex is real-rooted and that the antiprism triangulation of ∆ has the almost strong Lefschetz property over R for every shellable complex ∆. Several related open problems are discussed.