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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 8 Pith papers
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Matroid flat counts can have many peaks
For every positive integer m there is a matroid whose rank-by-rank counts of flats have exactly m peaks, disproving Rota's unimodality conjecture and Mason's log-concavity conjecture.
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Kazhdan-Lusztig polynomials of matroids need not be unimodal
Deleting carefully chosen points from finite projective geometries yields representable matroids whose Kazhdan–Lusztig polynomials are not unimodal, so they need not be log-concave or real-rooted.
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An Intersection Product for the Polytope Algebra
A new 'intersection product' on the polytope algebra is constructed, and its volumetric analogue of the graded Möbius algebra is shown to satisfy hard Lefschetz and Hodge-Riemann in degree one.
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Koszul Graded M\"obius Algebras and Strongly Chordal Graphs
For cycle matroids of graphs, the graded Möbius algebra is Koszul if and only if the graph is strongly chordal, with a new edge-ordering characterization of strong chordality.
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The external activity complex of a pair of matroids
The paper proves Speyer's 2005 tropical f-vector conjecture by showing that the matroid invariant ω(M) is non-negative for every matroid, via a Cohen-Macaulay external activity complex of a pair of matroids.
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The inverse $Z$-polynomial of a matroid
Explicit formulas for the inverse Z-polynomial of uniform and sparse paving matroids are derived, and the coefficients are shown to be unimodal and log-concave for sparse paving matroids.
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Log-concavity of inverse Kazhdan-Lusztig polynomials of paving matroids
The inverse Kazhdan-Lusztig polynomials of paving matroids are log-concave, proved via a new real-rootedness result for a Hadamard product with (1+t)^n.
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Wonderful Compactification of a Cartan Subalgebra of a Semisimple Lie Algebra
A Cartan subalgebra admits a wonderful compactification whose boundary components, affine paving, and cohomology are governed by the root system and its Coxeter arrangement.
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