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.
@article{10_37236_7105,
author = {Nicholas Proudfoot and Yuan Xu and Ben Young},
title = {The {\(Z\)-polynomial} of a matroid},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {1},
doi = {10.37236/7105},
zbl = {1380.05022},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7105/}
}
TY - JOUR
AU - Nicholas Proudfoot
AU - Yuan Xu
AU - Ben Young
TI - The \(Z\)-polynomial of a matroid
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/7105/
DO - 10.37236/7105
ID - 10_37236_7105
ER -
%0 Journal Article
%A Nicholas Proudfoot
%A Yuan Xu
%A Ben Young
%T The \(Z\)-polynomial of a matroid
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/7105/
%R 10.37236/7105
%F 10_37236_7105
Nicholas Proudfoot; Yuan Xu; Ben Young. The \(Z\)-polynomial of a matroid. The electronic journal of combinatorics, Tome 25 (2018) no. 1. doi: 10.37236/7105
