Séminaire Lotharingien de Combinatoire, 85B.29 (2021), 12 pp.
Max Hlavacek and Liam Solus
Subdivisions of Shellable Complexes
Abstract.
This extended abstract is a summary of a recent paper
which studies the enumeration of faces of subdivisions of cell complexes.
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 polytopal complexes by relating interlacing polynomials to shellability via the existence of so-called stable shellings.
We show that any shellable cubical, or simplicial, complex admitting a stable shelling has barycentric and edgewise subdivisions with real-rooted h-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.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
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