Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, II: Groups of type F, G, and H

Abstract

This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as rationality and functional equations. Here, we calculate such bivariate zeta functions of three infinite families of nilpotent groups of class 2 generalizing the Heisenberg group of ([Formula: see text])-unitriangular matrices over rings of integers of number fields. The local factors of these zeta functions are also expressed in terms of sums over finite hyperoctahedral groups, which provide formulae for joint distributions of three statistics on such groups.

Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, I: Arithmetic properties

“…It would be interesting to understand which other kinds of information one can extract from bivariate zeta functions. In [11], we explicitly compute bivariate zeta functions of three infinite families of nilpotent groups. As a consequence, we obtain explicit formulae for two (univariate) zeta functions of these groups.…”

“…As a consequence, we obtain explicit formulae for two (univariate) zeta functions of these groups. We also provide an application in combinatorics: the formulae for bivariate representation zeta functions of these groups are shown to be related to statistics of certain Weyl groups, leading to formulae for joint distributions of three statistics; see [11,Propositions 5.5 and 5.6].…”

Enumerating conjugacy classes of graphical groups over finite fields

“…The class‐counting zeta function of $$\mathbf {G}$$ is the Dirichlet series $$\zeta ^{\operatorname{k}}_{\mathbf {G}}(s) = \sum _{i=0}^\infty \operatorname{k}(\mathbf {G}(\mathfrak {O}/\mathfrak {P}^i)) {\vert \mathfrak {O}/\mathfrak {P}^i\vert} ^{-s}$$. Beginning with work of du Sautoy [7], these and closely related series enumerating conjugacy classes have recently been studied, see [3, 19, 20, 27–29]. Recall the definition of the conjugacy class zeta function $$\zeta ^{\rm cc}_G(s)$$ associated with a finite group $$G$$ from Section 4.…”

Linear relations with disjoint supports and average sizes of kernels

“…\end{equation*}In the literature, these and related functions are also called ‘conjugacy class’ and ‘class number’ zeta functions. For recent work in the area, see [3, 19, 20, 26, 28, 30].…”