Combinatorics Seminar: "Subdivisions of Shellable Complexes"

Abstract: In geometric combinatorics, we often study the unimodality of combinatorial generating polynomials, or the stronger property of the polynomial being real-rooted. Motivated by a conjecture of Brenti and Welker on the real-rootedness of the h-polynomial of the barycentric subdivision of the boundary complex of a convex polytope, we introduce a framework for proving real-rootedness of h-polynomials for subdivisions of certain polytopal complexes by relating interlacing polynomials to shellability via the existence of stable shellings. We show that any stably shellable cubical, or simplicial, complex admitting a stable shelling has barycentric and edgewise (when well-defined) subdivisions with real-rooted -polynomials. Such shellings are shown to exist for well-studied families of cubical polytopes, giving a positive answer to the conjecture of Brenti and Welker in these cases. The framework of stable shellings is also applied to answer to a conjecture of Mohammadi and Welker on edgewise subdivisions in the case of shellable simplicial complexes.

Host: Laura Escobar Vega