Randomized trial demonstrates Hodge-Riemann relations in matroids, suggesting advancements in combinatorial geometry.
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.
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Braden et al. (2026) studied this question.
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