Stressed Hyperplanes and Kazhdan–Lusztig Gamma-Positivity for Matroids

Abstract

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 that 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 $\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.