The Poincaré-extended a b $\mathbf {a}\mathbf {b}$ -index

Abstract

Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended $ab$-index, which generalizes both the $ab$-index and the Poincaré 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 Kühne. 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 — pose a conjecture about Schur positivity. This conjecture was strengthened and proved by Ricky Liu, and the proof appears as an appendix.