Geometric Realization of $\gamma$-Vectors of Subdivided Cross Polytopes

Abstract

For any flag simplicial complex $\Theta$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $\Delta(\Theta)$ whose $f$-vector is the $\gamma$-vector of $\Theta$. This proves that the $\gamma$-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.