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 or the subdivision of the boundary complex of a simple polytope. 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.
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F. Brenti and V. Welker are partially supported by EU Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272 and the program on “Algebraic Combinatorics” at the Mittag-Leffler Institut in Spring 2005.
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Brenti, F., Welker, V. f-Vectors of barycentric subdivisions. Math. Z. 259, 849–865 (2008). https://doi.org/10.1007/s00209-007-0251-z
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DOI: https://doi.org/10.1007/s00209-007-0251-z