Irreducible Representations of Baumslag-Solitar Groups

TL;DR

This paper classifies all finite-dimensional irreducible linear representations of Baumslag-Solitar groups BS(p,q) for coprime p and q by analyzing the Zariski closure of their matrix images.

Contribution

It provides a complete classification of irreducible representations of BS(p,q) for coprime p and q, using algebraic group techniques.

Findings

Explicit description of irreducible representations

Connection between representations and algebraic group closures

Framework applicable to similar group classification problems

Abstract

We classify the finite-dimensional irreducible linear representations of the Baumslag-Solitar groups BS(p,q) = < a, b | a b^p = b^q a > for relatively prime p and q. The general strategy of the argument is to consider the matrix group given by image of a representation and study its Zariski closure in GL(n, C).