Existence of Kähler algebras with Chow polynomials as Hilbert series

In this article, we study Chow polynomials of weakly ranked posets and prove the existence of Gorenstein algebras with the Kähler package such that their Hilbert--Poincaré series agrees with the Chow polynomial. Our statement provides evidence in support of a conjecture by Ferroni, Matherne and the second author about the existence of an algebra for every weakly ranked poset that generalizes the Feichtner--Yuzvinsky Chow ring for matroids. This allows us to prove strong inequalities for the coefficients of Chow polynomials; we prove log-concavity for all posets of weak rank at most six and provide counterexamples to log-concavity for any higher rank. For ranked posets we recover an even stronger condition, showing that the differences between consecutive coefficients constitute a pure O-sequence.