Adam Schweitzer: Existence of Hard Lefschetz algebras with Chow polynomials as Hilbert series

Abstract: In recent years, the introduction of Chow rings of matroids has led to breakthroughs in matroid theory, including proofs of the Heron--Rota--Welsh and Mason conjectures.

Chow polynomials of posets generalize the Hilbert--Poincaré polynomial of the Chow ring of matroids. While no such ring-theoretic interpretation is known for general posets, several properties surprisingly still hold (including positivity, unimodality and palindromicity). This observation has led to the conjecture that Chow rings should generalize to arbitrary posets, and (among other requirements) satisfy the Hard Lefschetz property.

As a step towards this conjecture, we show that the Chow polynomial of any poset satisfies the numerical constraints imposed by the Hilbert series of Hard Lefschetz algebras. Consequently, there exists an algebra with the Hard Lefschetz property (and even the full Kähler package) that has the Chow polynomial of any poset as its Hilbert--Poincaré polynomial. Equivalently, this shows that the coefficients of the Chow polynomial are the h-vector of a simple polytope.

This is a joint work with Lorenzo Vecchi.