The $Z$-Polynomial of a Matroid

  • Nicholas Proudfoot
  • Yuan Xu
  • Ben Young
Keywords: matroids, Kazhdan-Lusztig polynomials


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.

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