# <   AlCoVE: an Algebraic Combinatorics Virtual Expedition   >

## Martina Juhnke-Kubitzke

The antiprism triangulation provides a natural way to subdivide a simplicial complex $$\Delta$$, 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 $$\Delta$$, from a combinatorial point of view, and by iterated crossings of the faces of $$\Delta$$, from a geometric point of view.

This talk studies enumerative invariants and algebraic properties associated with this triangulation, such as the transformation of the $$h$$-vector of $$\Delta$$ under antiprism triangulation. The main results show that the $$h$$-polynomial of the antiprism triangulation of a simplex is real-rooted and that the antiprism triangulation of $$\Delta$$ has the almost strong Lefschetz property over $$\mathbb{R}$$ for every shellable complex $$\Delta$$. This is joint work with Christos Athanasiadis and Jan-Marten Brunink.