Geometric realization of γ-vectors of subdivided cross polytopes

Abstract

For any flag simplicial complex Θ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex ∆(Θ) whose f-vector is the γ-vector of Θ. This proves that the γ-vector of any such simplicial complex is the face vector of a flag simplicial complex, partially solving a conjecture by Nevo and Petersen. As a corollary we obtain that such simplicial complexes satisfy the Frankl-Füredi-Kalai inequalities.