Generalized Igusa functions and ideal growth in nilpotent Lie rings

Angela Carnevale, Michael M. Schein, Christopher Voll

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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

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


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