A Flip-Free Proof of Log-Concavity for Matroids
Spencer Backman, Ph.D.
University of Vermont
Monday, September 9th, 4:00 PM
In 2015 Adiprasito, Huh, and Katz settled the famous Heron-Rota-Welsh conjecture that the absolute value of the coefficients of the characteristic polynomial of a matroid are log-concave. The approach of AHK was to show that these coefficients can be interpreted as intersection numbers in the Chow ring of a matroid previously introduced by Feichtner and Yuzvinsky. They then establish a K\"ahler package for the Chow ring of a matroid: Poincar\'e duality, the Hard Lefschetz theorem, and the Hodge-Riemann relations, and show that the desired log-concavity follows from the degree 1 part of the Hodge-Riemann relations (the Hodge index theorem). The AHK proof of the K\"ahler package for matroids is inspired by earlier work of McMullen on simple polytopes and thus utilizes a notion of "flipping" which provides a fine interpolation between matroids and projective space. For achieving their goal, AHK prove that the K\"ahler package respects flipping. This impressive program comes at a cost of working with more general objects than matroids which can obscure some combinatorial and geometric information. I will describe joint work with Chris Eur and Connor Simpson where we introduce a new presentation for the Chow ring of a matroid and apply this presentation to obtain a new proof of the Hodge index theorem for matroids which eschews the use of flipping and thus does not leave the realm of matroids.