Elena Hoster: Chow polynomials of uniform matroids

Abstract: The Chow polynomial of a matroid is the Hilbert series of its Chow ring, defined using the maximal building set. This polynomial satisfies several combinatorial properties: it is palindromic, unimodal, log-concave, and gamma-positive. One proof of its gamma-positivity, due to Stump, arises from its connection to the Poincaré-extended ab-index, a polynomial introduced by Dorpalen-Barry, Maglione, and Stump.
With the resulting gamma-expansion, we derive an explicit combinatorial formula for the Chow polynomial of uniform matroids. Finally, we discuss conjectures related to these structures, including the question of their real-rootedness.