The Poincaré-extended 𝐚𝐛-index
Dorpalen-Barry, Galen Maglione, Joshua Stump, Christian
Dorpalen-Barry, Galen
Maglione, Joshua
Stump, Christian
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Publication Date
2024-12-20
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journal article
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Dorpalen-Barry, Galen, Maglione, Joshua, & Stump, Christian. (2025). The Poincaré-extended ab-index. Journal of the London Mathematical Society, 111(1), e70054. doi:https://doi.org/10.1112/jlms.70054
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.
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Publisher
Wiley and London Mathematical Society
Publisher DOI
https://doi.org/10.1112/jlms.70054
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Attribution 4.0 International