The γ\gamma-vector of a barycentric subdivision

We prove that the $\gamma$-vector of the barycentric subdivision of a simplicial sphere is the $f$-vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the $h$-vector of the barycentric subdivision of a boolean complex

Gamma-positivity in combinatorics and geometry

Gamma-positivity is an elementary property that polynomials with symmetric coefficients may have, which directly implies their unimodality. The idea behind it stems from work of Foata, Sch\"utzenberger and Strehl on the Eulerian polynomials; it was revived independently by Br\"and\'en and Gal in the course of their study of poset Eulerian polynomials and face enumeration of flag simplicial spheres, respectively, and has found numerous applications since then. This paper surveys some of the main results and open problems on gamma-positivity, appearing in various combinatorial or geometric contexts, as well as some of the diverse methods that have been used to prove it.Comment: Final versio

Combinatorics of antiprism triangulations

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 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 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 $\Delta$ has the almost strong Lefschetz property over ${\mathbb R}$ for every shellable complex $\Delta$. Several related open problems are discussed.Comment: 33 pages, 2 figure