Hodge theory for combinatorial geometries

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 $\mathrm {M}$. 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.

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    @INCOLLECTION{Zaslavsky,
      author = {Zaslavsky, Thomas},
      title = {The {M}öbius function and the characteristic polynomial},
      booktitle = {Combinatorial Geometries},
      series = {Encyclopedia Math. Appl.},
      volume = {29},
      pages = {114--138},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {1987},
      mrclass = {03B35 (05A15 05C15)},
      mrnumber = {0921071},
      zblnumber = {0632.05017},
      doi = {10.1017/CBO9781107325715.009},
      }

Authors

Karim Adiprasito

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

June Huh

Institute for Advanced Study, Princeton, NJ, USA and Korea Institute for Advanced Study, Seoul, Korea

Eric Katz

Department of Mathematics, The Ohio State University, Columbus, Ohio, USA