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The \(Z\)-polynomial of a matroid. (English) Zbl 1380.05022
Summary: We introduce the \(Z\)-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the \(Z\)-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion, obtaining a closed formula for Kazhdan-Lusztig coefficients as alternating sums of multi-indexed Whitney numbers. For realizable matroids, we give a cohomological interpretation of the \(Z\)-polynomial in which the symmetry is a manifestation of Poincaré duality.

MSC:
05B35 Combinatorial aspects of matroids and geometric lattices
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
55N33 Intersection homology and cohomology in algebraic topology
11B83 Special sequences and polynomials
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
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