Stressed hyperplanes and Kazhdan-Lusztig gamma-positivity for matroids

TL;DR

This paper introduces stressed hyperplanes in matroids, providing concise formulas for key polynomials of paving matroids, and explores their gamma-positivity with combinatorial interpretations, advancing understanding of matroid invariants.

Contribution

It defines stressed hyperplanes, derives formulas for Kazhdan--Lusztig and related polynomials, and studies gamma-positivity with explicit combinatorial interpretations.

Findings

Concise formulas for polynomials of paving matroids

Gamma-positivity with positive coefficients in many cases

Combinatorial interpretations via Young tableaux

Abstract

In this article we make several contributions of independent interest. First, we introduce the notion of stressed hyperplane of a matroid, essentially a type of cyclic flat that permits to transition from a given matroid into another with more bases. Second, we prove that the framework provided by the stressed hyperplanes allows one to write very concise closed formulas for the Kazhdan--Lusztig, inverse Kazhdan--Lusztig and $Z$-polynomials of all paving matroids, a class which is conjectured to predominate among matroids. Third, noticing the palindromicity of the $Z$-polynomial, we address its $\gamma$-positivity, a midpoint between unimodality and real-rootedness. To this end, we introduce the \emph{$\gamma$-polynomial} associated to it, we study some of its basic properties and we find closed expressions for it in the case of paving matroids. Also, we prove that it has positive coefficients in many interesting cases, particularly in the large family of sparse paving matroids, and other smaller classes such as projective geometries, thagomizer matroids and other particular graphs. Our last contribution consists of providing explicit combinatorial interpretations for the coefficients of many of the polynomials addressed in this article by enumerating fillings in certain Young tableaux and skew Young tableaux.