Macedo Lins de Araujo, Paula (2019) Bivariate representation and conjugacy classzeta functions associated to unipotent groupschemes. PhD thesis, Bielefeld University.
Full content URL: https://doi.org/10.4119/unibi/2933600
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Item Type:  Thesis (PhD) 

Item Status:  Live Archive 
Abstract
The main topic of this doctoral thesis is zeta functions of groups. Let G be a unipotent group scheme defined over the ring of integers O of a number field. The group G(O) of Orational points is a finitely generated torsionfree nilpotent group. We introduce two bivariate zeta functions related to groups of the form G(O): firstly the bivariate representation zeta function of G(O), which enumerates the isomorphism classes of irreducible complex representations of finite dimensions of its congruence quotients, and secondly the bivariate conjugacy class zeta function of G(O), which enumerates the conjugacy classes of each size of its congruence quotients. These zeta functions might be used as tools for understanding another (univariate) zeta functions, as they both specialise to class number zeta functions, which enumerate class numbers of the congruence quotients. Additionally, in case of nilpotency class two, bivariate representation zeta functions specialise to twist representation zeta functions, which are zeta functions enumerating the irreducible complex characters of finite dimensions up to tensoring by onedimensional characters. We show that bivariate representation and bivariate conjugacy class zeta functions satisfy Euler decompositions and that almost all of their Euler factors are rational and satisfy functional equations. We also prove that they converge on some domains of C^2 and, furthermore, their maximal domains of convergence and meromorphic continuation are independent of the number field O considered, up to finitely many local factors. We provide formulae for the bivariate zeta functions of three infinite families of groups of nilpotency class 2 of the form G(O) which generalise the Heisenberg group of 3×3unitriangular matrices over O. As an application, we establish formulae for the joint distributions of three statistics on finite hyperoctahedral groups.
Keywords:  Group theory, Zeta functions, Finitely generated nilpotent groups, Conjugacy classes, Irreducible complex characters, pAdic integration, Signed permutation statistics 

Subjects:  G Mathematical and Computer Sciences > G110 Pure Mathematics 
Divisions:  College of Science > School of Mathematics and Physics 
ID Code:  47834 
Deposited On:  29 Mar 2022 15:37 
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