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Singular Hodge theory for combinatorial geometries
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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.
Forward citations
Cited by 3 Pith papers
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Polar Degrees of Matroids
Polar degrees of reciprocal linear spaces are given by coefficients of a substitution of the matroid's reduced characteristic polynomial.
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Equivariant gamma-positivity of matroid Chow rings
Proves equivariant gamma-positivity for matroid Chow rings, giving new combinatorial interpretations and confirming a conjecture.
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Inverse Kazhdan--Lusztig polynomials of fan matroids
Explicit formulas and generating functions are obtained for inverse Kazhdan-Lusztig and inverse Z-polynomials of fan matroids, with coefficients proven log-concave and internally zero-free.
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