Irreducible representations of a family of groups of maximal nilpotency class I: the non-exceptional case

TL;DR

This paper develops a constructive method to compute p-local representation zeta functions for a family of finitely generated nilpotent groups with maximal nilpotency class, focusing on irreducible representations of certain dimensions.

Contribution

It introduces a standard form for matrices of irreducible representations and counts twist isoclasses, enabling the calculation of p-local zeta functions for most cases.

Findings

Computed p-local zeta functions for all but finitely many cases

Established a standard form for irreducible representations

Provided explicit counts of twist isoclasses

Abstract

We use a constructive method to obtain all but finitely many p-local representation zeta functions of a family Mn of finitely generated nilpotent groups with maximal nilpotency class. For representation dimensions coprime to all primes p < n, we construct all irreducible representations of Mn by defining a standard form for the matrices of these representations and, after taking into account twisting and isomorphism, count these twist isoclasses to obtain our p-local zeta functions.