Lorenzo Vecchi: Chow polynomials for arbitrary building sets, uniform matroids and the Deligne–Knudsen–Mumford moduli space

Abstract: 

We study the properties and the roots of the Chow polynomial H, a variation of the chain polynomial of a poset, equipped with some combinatorial data called building set. Its name originates from the fact that when the poset is a geometric lattice and the building set is maximal, this is the Hilbert–Poincaré series of the Chow ring of the matroid. However, when the matroid is a complete graph and the building set is minimal, one recover exactly the cohomology of the Deligne–Knudsen–Mumford moduli space, $\overline{M}_{0,n+1}$. Enticingly, in both cases the Chow polynomial is conjectured to be real-rooted.

In this talk we prove real-rootedness of H in the uniform case and introduce new formulas to compute it for arbitrary building sets. We show how our techniques let us recover known formulas for the complete graph and discuss some counterexamples to log-concavity for arbitrary building sets.

This is joint work with Petter Brändén and ongoing work with Christopher Eur, Luis Ferroni, Jacob Matherne and Roberto Pagaria.