Mathieu's approach to the Jacobian Conjecture

TL;DR

This paper explains Mathieu's Lie group approach to the Jacobian Conjecture, highlighting how a group-theoretic conjecture implies it, with detailed background and proof exposition.

Contribution

It provides an accessible exposition of Mathieu's proof linking Lie group theory to the Jacobian Conjecture, including new insights on representation theory.

Findings

Mathieu's proof connects Lie group conjecture to the Jacobian Conjecture.

Representation theory results on $SL(N,\mathbb{C})$ subrepresentations.

Expository review of Mathieu's approach and proof.

Abstract

In this paper, we give an expository presentation of the paper of Olivier Mathieu. The paper of Mathieu proves that a Lie group-theoretic conjecture implies the Jacobian Conjecture. To give Mathieu's proof, we first review the required literature on representation theory in an expository way. We continue to prove some results on the irreducible subrepresentations of the tensor algebra of the standard representation of $SL(N,\mathbb{C})$. The last part of the paper is dedicated to Mathieu's proof.