Séminaire Lotharingien de Combinatoire, 84B.71 (2020), 12 pp.

Angela Carnevale, Michael M. Schein and Christopher Voll

Generalized Igusa Functions and Ideal Growth in Nilpotent Lie Rings

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.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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