Kazhdan-Lusztig polynomials of matroids under deletion

TL;DR

This paper derives a formula connecting the Kazhdan-Lusztig polynomial of a matroid to those of related matroids obtained by deletion, contraction, and localization, with applications to graphic matroids.

Contribution

It introduces a new formula relating Kazhdan-Lusztig polynomials of matroids under deletion and other operations, expanding understanding of their combinatorial properties.

Findings

Derived a formula linking Kazhdan-Lusztig polynomials of matroids and their deletions

Applied the formula to compute polynomials for graphic matroids

Provided a simple formula for parallel connection graphs

Abstract

We present a formula which relates the Kazhdan-Lusztig polynomial of a matroid $M$, as defined by Elias, Proudfoot and Wakefield, to the Kazhdan--Lusztig polynomials of the matroid obtained by deleting an element, and various contractions and localizations of $M$. We give a number of applications of our formula to Kazhdan--Lusztig polynomials of graphic matroids, including a simple formula for the Kazhdan--Lusztig polynomial of a parallel connection graph.