Flag subdivisions and $γ$-vectors

Details Flag subdivisions and $γ$-vectors Flag subdivisions and $γ$-vectors

2011-06-22

arxiv.org/abs/1106.4520v4

Mathematics

Combinatorics

Substantial revision, with new results and references added

Scientific paper

The $\gamma$-vector is an important enumerative invariant of a flag simplicial homology sphere. It has been conjectured by Gal that this vector is nonnegative for every such sphere $\Delta$ and by Reiner, Postnikov and Williams that it increases when $\Delta$ is replaced by any flag simplicial homology sphere which geometrically subdivides $\Delta$. Using the nonnegativity of the $\gamma$-vector in dimension 3, proved by Davis and Okun, as well as Stanley's theory of simplicial subdivisions and local $h$-vectors, the latter conjecture is confirmed in this paper in dimensions 3 and 4.

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