# The Z-polynomial of a matroid

## TL;DR

This paper introduces the Z-polynomial for matroids, explores its symmetry, derives a recursion for Kazhdan-Lusztig coefficients, and provides a closed-form formula with cohomological interpretation for realizable cases.

## Contribution

It defines the Z-polynomial, uncovers its symmetry, and derives a new recursion and closed formula for Kazhdan-Lusztig coefficients in matroids.

## Key findings

- Derived a recursion for Kazhdan-Lusztig coefficients
- Obtained a closed formula as alternating sums of Whitney numbers
- Provided cohomological interpretation for realizable matroids

## Abstract

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 Poincare duality.