Simplicial generation of Chow rings of matroids

  • Spencer Backman

    The University of Vermont, Burlington, USA
  • Christopher Eur

    Harvard University, Cambridge, USA
  • Connor Simpson

    University of Wisconsin - Madison, USA
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We introduce a presentation of the Chow ring of a matroid by a new set of generators, called “simplicial generators.” These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation via the theory of matroid quotients. Using this combinatorial interpretation, we (i) produce a bijection between a monomial basis of the Chow ring and a relative generalization of Schubert matroids, (ii) recover the Poincaré duality property, (iii) give a formula for the volume polynomial, which we show is log-concave in the positive orthant, and (iv) recover the validity of Hodge–Riemann relations in degree 1, which is the part of the Hodge theory of matroids that currently accounts for all combinatorial applications of the work of Adiprasito et al. (2018). Our work avoids the use of “flips,” the key technical tool employed in that work.

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Spencer Backman, Christopher Eur, Connor Simpson, Simplicial generation of Chow rings of matroids. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1350