Colloquium: Combinatorics and Positivity of Chow Polynomials of Matroids

Abstract: Matroids are combinatorial structures that capture the notion of independence and provide a common framework for studying objects such as graphs and vector configurations. Chow rings of matroids, introduced by Feichtner and Yuzvinsky in 2004, are algebraic structures that encode rich combinatorial information about matroids. In recent years, they have played a central role in the resolution of several longstanding conjectures concerning inequalities for matroid invariants, through the work of June Huh and his collaborators.

In this talk, I will explore the connections between the algebraic and combinatorial aspects of matroid Chow rings, with a particular focus on their Hilbert series, also known as Chow polynomials. I will present combinatorial formulas for the Chow polynomials of uniform matroids and their q-analogs in terms of permutations and permutation statistics. These formulas lead to surprising positivity properties that extend beyond uniform matroids to general matroids. I will also discuss how these positivity phenomena can be refined to incorporate symmetries, leading to equivariant gamma-positivity. If time permits, I will conclude with some recent developments and open directions in this area.