An explicit embedding of the Tits group in the compact real form of $E_6$

TL;DR

This paper explicitly constructs a set of 27x27 unitary matrices to embed the Tits group into the compact real form of the E6 Lie group, also embedding PSL_2(25) into F4.

Contribution

It provides the first explicit matrix-based embedding of the Tits group into the compact real form of E6, and embeds PSL_2(25) into F4.

Findings

Explicit 27x27 unitary matrices for Tits group embedding

Embedding of PSL_2(25) into F4 via subset matrices

Concrete realization of complex Lie group embeddings

Abstract

We construct a set of $27\times 27$ unitary matrices which give an explicit embedding of the Tits group in the compact real form of the Lie group of type $E_6$. A subset gives an embedding of $\mathrm{PSL}_2(25)$ in $F_4$.