On matroid modularity and the coefficients of the inverse Kazhdan-Lusztig polynomial of a matroid

TL;DR

This paper explores properties of the inverse Kazhdan-Lusztig polynomial of matroids, linking it to matroid degeneracy and proving the conjecture for modular matroids.

Contribution

It establishes new connections between matroid degeneracy and inverse Kazhdan-Lusztig polynomials, proving the conjecture for modular matroids.

Findings

Degenerate modular matroids are not regular.

The conjecture holds for modular matroids.

Properties of inverse Kazhdan-Lusztig polynomials are characterized.

Abstract

Following the work of Gao and Xie in [2], we state some properties of the inverse Kazhdan-Lusztig polynomial of a matroid. We also give partial answers to a conjecture that states that regular connected matroids are non-degenerate. We link the degeneracy of a matroid to the inverse Kazhdan-Lusztig polynomial and we show that the Conjecture holds for modular matroids, by proving that degenerate modular matroids are not regular.