$f$-Vectors of Barycentric Subdivisions

Abstract

For a simplicial complex or more generally Boolean cell complex $Δ$ we study the behavior of the $f$- and $h$-vector under barycentric subdivision. We show that if $Δ$ has a non-negative $h$-vector then the $h$-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version of the Charney-Davis conjecture for spheres that are the subdivision of a Boolean cell complex. For a general $(d-1)$-dimensional simplicial complex $Δ$ the $h$-polynomial of its $n$-th iterated subdivision shows convergent behavior. More precisely, we show that among the zeros of this $h$-polynomial there is one converging to infinity and the other $d-1$ converge to a set of $d-1$ real numbers which only depends on $d$.