Generalized Igusa functions and ideal growth in nilpotent Lie rings

Angela Carnevale, Michael M. Schein, Christopher Voll

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Abstract

We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new rational functions, and thus also the local zeta functions under consideration, enjoy a self-reciprocity property, expressed in terms of a functional equation upon inversion of variables. We establish a conjecture of Grunewald, Segal, and Smith on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups.

Original languageEnglish
Article number71
JournalSeminaire Lotharingien de Combinatoire
Issue number84
StatePublished - 2020

Bibliographical note

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Funding

[email protected]. Partially supported by the Irish Research Council (GOIPD/2018/319). †[email protected]. ‡[email protected]. Partially supported by The Emmy Noether Minerva Research Institute at Bar-Ilan University during the preliminary stages of this project The research of all three authors was supported by a grant from the GIF, the German– Israeli Foundation for Scientific Research and Development (1246/2014)

FundersFunder number
Emmy Noether Minerva Research Institute at Bar-Ilan University
German-Israeli Foundation for Scientific Research and Development1246/2014
Irish Research CouncilGOIPD/2018/319

    Keywords

    • Igusa functions
    • combinatorial reciprocity theorem
    • normal zeta functions
    • subgroup growth
    • weak orders

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