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Singular Hodge theory for combinatorial geometries (2010.06088v4)
Published 13 Oct 2020 in math.CO and math.AG
Abstract: We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e 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.
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