Inverse Kazhdan–Lusztig polynomials of matroids under deletion

Abstract

We provide a deletion formula for the inverse Kazhdan–Lusztig polynomial and the inverse $Z$-polynomial of a matroid. Our formulas provide analogues to the deletion formulas of Braden–Vysogorets for Kazhdan–Lusztig and $Z$-polynomials. We discuss several consequences, which include closed formulas and recursions for these invariants on uniform matroids, projective geometries, glued cycles, and arbitrary matroids of corank $2$. As a relevant application of our deletion formula, we show the existence of a matroid of rank $19$ which disproves a conjecture of Xie and Zhang concerning a real-rootedness property of inverse Kazhdan–Lusztig polynomials.