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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