Equivariant Kazhdan-Lusztig theory of paving matroids

TL;DR

This paper develops a method to compute equivariant Kazhdan-Lusztig polynomials for paving matroids, including those with symmetries from Mathieu groups, by analyzing their behavior under relaxation of stressed hyperplanes.

Contribution

It introduces a new approach to calculate equivariant polynomials for paving matroids, extending the understanding of their algebraic and combinatorial properties.

Findings

Computed equivariant Kazhdan-Lusztig polynomials for various paving matroids

Analyzed the effect of relaxation of stressed hyperplanes on these polynomials

Applied the method to matroids with Mathieu group symmetries

Abstract

We study the way in which equivariant Kazhdan-Lusztig polynomials, equivariant inverse Kazhdan-Lusztig polynomials, and equivariant Z-polynomials of matroids change under the operation of relaxation of a collection of stressed hyperplanes. This allows us to compute these polynomials for arbitrary paving matroids, which we do in a number of examples, including various matroids associated with Steiner systems that admit actions of Mathieu groups.