Existence of K\"ahler algebras with Chow polynomials as Hilbert series

TL;DR

This paper demonstrates the existence of Gorenstein algebras with Hilbert series matching Chow polynomials of weakly ranked posets, supporting a conjecture and revealing new inequalities and properties of these polynomials.

Contribution

It proves the existence of Kähler algebras with specified Hilbert series for weakly ranked posets, extending the Feichtner–Yuzvinsky Chow ring and establishing new inequalities.

Findings

Proves existence of Gorenstein algebras matching Chow polynomials.

Establishes log-concavity for posets of rank ≤ 6.

Provides counterexamples to log-concavity for higher ranks.

Abstract

In this article, we study Chow polynomials of weakly ranked posets and prove the existence of Gorenstein algebras with the K\"ahler package such that their Hilbert--Poincar\'e 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.