The Poincar\'e-extended ab-index

TL;DR

This paper introduces the Poincaré-extended ab-index, a new combinatorial invariant that generalizes existing indices, proves nonnegativity for certain posets, and confirms conjectures related to hyperplane arrangements and quasisymmetric functions.

Contribution

It defines the Poincaré-extended ab-index, provides a combinatorial description for R-labeled posets, and proves conjectures for hyperplane arrangement intersection posets.

Findings

Proves nonnegativity of coefficients for R-labeled posets.

Confirms conjectures for intersection posets of hyperplane arrangements.

Establishes connections to quasisymmetric functions and Schur positivity.

Abstract

Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincar\'e-extended ab-index, which generalizes both the ab-index and the Poincar\'e polynomial. For posets admitting R-labelings, we give a combinatorial description of the coefficients of the extended ab-index, proving their nonnegativity. In the case of intersection posets of hyperplane arrangements, we prove the above conjecture of the second author and Voll as well as another conjecture of the second author and K\"uhne. We also define the pullback ab-index generalizing the cd-index of face posets for oriented matroids. Our results recover, generalize and unify results from Billera-Ehrenborg-Readdy, Bergeron-Mykytiuk-Sottile-van Willigenburg, Saliola-Thomas, and Ehrenborg. This connection allows us to translate our results into the language of quasisymmetric functions, and-in the special case of symmetric functions-make a conjecture about Schur positivity. A proof of this conjecture now appears an appendix by Ricky Ini Liu.