Abstract
We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the log-concavity of the coefficients of the characteristic polynomial of M. We furthermore conclude that the $f$-vector of the independence complex of a matroid forms a log-concave sequence, proving a conjecture of Mason and Welsh for general matroids.
Citation
Kareem Adiprasito. June Huh. Eric Katz. "Hodge theory for combinatorial geometries." Ann. of Math. (2) 188 (2) 381 - 452, September 2018. https://doi.org/10.4007/annals.2018.188.2.1
Information