On $\gamma$-vectors satisfying the Kruskal-Katona inequalities

Abstract

We present examples of flag homology spheres whose $\gamma$-vectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit simplicial complexes whose $f$-vectors are the $\gamma$-vectors in question. In another direction, we show that if a flag $(d-1)$-sphere has at most $2d+2$ vertices its $\gamma$-vector satisfies the Kruskal-Katona inequalities. We conjecture that if $\Delta$ is a flag homology sphere then $\gamma (\Delta )$ satisfies the Kruskal-Katona inequalities. This conjecture is a significant refinement of Gal's conjecture, which asserts that such $\gamma$-vectors are nonnegative.

18 pages; Our main result and conjectures have been strengthened. Also we now have explicit constructions of simplicial complexes whose $f$-vectors are the $\gamma$-vectors in question

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• Face enumeration on flag complexes and flag spheres
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