Lorenzo Vecchi: Singular cohomology for matroids and real-rootedness

Abstract:

The Chow ring and the augmented Chow ring of a matroid are graded rings associated with two combinatorial fans: the Bergman fan and the augmented Bergman fan. Their Hilbert series are always unimodal by the Strong Lefschetz property, and for many matroids, including uniform matroids, they are also real-rooted.

We consider related geometric invariants of these fans, called the singular cohomology ring and the augmented singular cohomology ring, and study their properties for uniform matroids. These rings may be viewed as generalizations of the corresponding Chow rings.

We investigate the zeros of their refined Hodge—Poincaré polynomials and their relationship with the Strong Lefschetz property. For the singular cohomology ring of a uniform matroid, the Strong Lefschetz property is known to hold, and we prove that the corresponding polynomials are real-rooted, resolving a conjecture of Binder.

In the augmented setting, the Strong Lefschetz property fails, as do both unimodality and real-rootedness. We construct a natural modification of the augmented Bergman fan that restores both the Strong Lefschetz property and the real-rootedness of the associated Hodge—Poincaré polynomials.

These results extend an earlier real-rootedness result for uniform matroids due to Petter Brändén and me. Our techniques include the deranged map of Brändén and Solus and Athanasiadis’ Eulerian transformation.

This is joint work with Kyle Binder.