On the conjecture of Kashuba and Mathieu about free Jordan algebras

TL;DR

This paper investigates Kashuba and Mathieu's conjecture on free Jordan algebras, providing computational evidence that refutes its validity despite strong supporting data.

Contribution

It offers new computational data on free Jordan algebras and demonstrates that the conjecture by Kashuba and Mathieu is false.

Findings

Computational data contradicts the conjecture

The conjecture does not hold despite positive evidence

Insights into the structure of free Jordan algebras

Abstract

Kashuba and Mathieu proposed a conjecture on vanishing of Lie algebra homology, implying a description of the $GL_d$-module structure of the free $d$-generated Jordan algebra. Their conjecture relies on a functorial version of the Tits--Kantor--Koecher construction that builds Lie algebras out of Jordan algebras. In this note, we summarize new intricate computational data concerning free Jordan algebras and explain why, despite a lot of overwhelmingly positive evidence, the conjecture of Kashuba and Mathieu is not true.