Matroid relaxations and Kazhdan-Lusztig non-degeneracy

TL;DR

This paper explores how circuit-hyperplane relaxation affects Kazhdan-Lusztig polynomials in matroids, providing formulas, bounds, and applications to non-degeneracy and sparse paving matroids.

Contribution

It introduces a family of rank-dependent polynomials linking matroid polynomials before and after relaxation, and proves non-degeneracy for matroids with a free basis.

Findings

Matroids with a free basis are non-degenerate.

Derived bounds and explicit formulas for polynomial coefficients.

Established relationships between matroid polynomials and relaxations.

Abstract

In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan--Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate the Kazhdan--Lusztig, the inverse Kazhdan--Lusztig and the $Z$-polynomial of each matroid with those of its relaxations. As an application of our main theorem, we prove that all matroids having a free basis are non-degenerate. Additionally, we obtain bounds and explicit formulas for all the coefficients of the Kazhdan--Lusztig, inverse Kazhdan--Lusztig and $Z$-polynomial of all sparse paving matroids.