Combinatorics of antiprism triangulations
Abstract
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 multipointed faces of $\Delta$, from a combinatorial point of view, and by successively applying the antiprism construction, or balanced stellar subdivisions, on the faces of $\Delta$, from a geometric point of view. This paper studies enumerative invariants associated to this triangulation, such as the transformation of the $h$vector of $\Delta$ under antiprism triangulation, and algebraic properties of its StanleyReisner ring. Among other results, it is shown that the $h$polynomial of the antiprism triangulation of a simplex is realrooted and that the antiprism triangulation of $\Delta$ has the almost strong Lefschetz property over ${\mathbb R}$ for every shellable complex $\Delta$. Several related open problems are discussed.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.10789
 Bibcode:
 2020arXiv200610789A
 Keywords:

 Mathematics  Combinatorics;
 05E45;
 05A05;
 05A18;
 13C14
 EPrint:
 Final version