Singular Hodge theory for combinatorial geometries Article Swipe

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matroid mathematics intersection homology hodge theory mathematical proof conjecture cohomology duality (order theory) pure mathematics intersection (aeronautics) riemann hypothesis algebra over a field discrete mathematics geometry aerospace engineering engineering

Tom Braden , June Huh , Jacob P. Matherne , Nicholas Proudfoot , Botong Wang ·

YOU? · · 2020 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2010.06088 · OA: W3093371845

We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy conjecture and the nonnegativity of the coefficients of Kazhdan-Lusztig polynomials for all matroids.