[Paper Review] Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes

de Araujo, Paula Macedo Lins|arXiv (Cornell University)|Feb 13, 2018

Advanced Combinatorial Mathematics22 references5 citations

TL;DR

This paper introduces bivariate zeta functions encoding the number of isomorphism classes of finite-dimensional irreducible complex representations and conjugacy classes in congruence quotients of unipotent group schemes over rings of integers in number fields. It establishes Euler factorization, rationality, and functional equations for almost all local factors, with explicit formulae for nilpotent groups of class two and joint distribution formulae via finite hyperoctahedral groups.

ABSTRACT

We introduce two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. These zeta functions encode, respectively, the numbers of isomorphism classes of irreducible complex representations of finite dimensions and the numbers of conjugacy classes of congruence quotients of the associated groups. Our bivariate zeta functions specialise to (univariate) class number zeta functions associated to the relevant groups. In case of nilpotency class two, bivariate representation zeta functions also specialise to (univariate) twist representation zeta functions. We show that these zeta functions satisfy Euler factorisations and almost all of their Euler factors satisfy rationality and functional equations. We give explicit formulae for the bivariate zeta functions of some families of nilpotent groups of class two. The local bivariate representation zeta functions of these groups are also expressed in terms of sums over finite hyperoctahedral groups, which provides formulae for joint distributions of three statistics on such groups.

Motivation & Objective

• To define bivariate zeta functions that encode representation and conjugacy class counts in congruence quotients of unipotent group schemes.
• To generalize univariate class number and twist representation zeta functions to a bivariate setting.
• To establish Euler factorization and rationality properties for local factors of these zeta functions.
• To derive explicit formulae for the zeta functions in the case of nilpotent groups of class two.
• To express local representation zeta functions in terms of sums over finite hyperoctahedral groups, yielding joint distribution formulae for three statistics.

Proposed method

• Define two bivariate zeta functions: one for irreducible representation counts and one for conjugacy class counts in finite quotients of unipotent group schemes.
• Specialize the bivariate functions to univariate class number and twist representation zeta functions in specific cases.
• Prove that the zeta functions admit Euler factorizations over the ring of integers of a number field.
• Show that almost all local Euler factors satisfy rationality and functional equations.
• Express local bivariate representation zeta functions as sums over finite hyperoctahedral groups.
• Derive explicit formulae for families of nilpotent groups of class two using these group-theoretic sums.

Experimental results

Research questions

• RQ1 How can bivariate zeta functions be constructed to simultaneously encode representation and conjugacy class data for unipotent group schemes?
• RQ2 Do these zeta functions satisfy Euler factorization and rationality properties for local factors?
• RQ3 In what way do the bivariate zeta functions specialize to known univariate zeta functions such as class number and twist representation zeta functions?
• RQ4 Can the local representation zeta functions be expressed in terms of symmetric group-like structures such as finite hyperoctahedral groups?
• RQ5 What joint distribution properties of three statistics on finite hyperoctahedral groups emerge from the zeta function expressions?

Key findings

• The bivariate zeta functions admit Euler factorization over the ring of integers of a number field.
• Almost all local Euler factors of the zeta functions satisfy rationality and functional equations.
• For nilpotent groups of class two, the bivariate representation zeta functions specialize to univariate twist representation zeta functions.
• Explicit formulae are derived for the bivariate zeta functions of families of nilpotent groups of class two.
• The local bivariate representation zeta functions are expressed as sums over finite hyperoctahedral groups, yielding formulae for the joint distribution of three statistics on these groups.
• The zeta functions provide a unified framework linking representation theory, conjugacy class counts, and group-theoretic statistics in unipotent group schemes.