Fixed points subgroups $G^{\sigma,\sigma'}$ by two involutive automorphisms $\sigma$, $\sigma'$ of exceptional compact Lie group $G$, Part II, $G = E_8$

TL;DR

This paper determines the structure of fixed point subgroups of the exceptional Lie group E_8 under two involutive automorphisms and describes the associated symmetric space, extending understanding of E_8's subgroup structure.

Contribution

It explicitly characterizes the subgroup $(E_8)^{\sigma,\sigma'}$ for two involutions and describes the corresponding symmetric space, providing concrete involutions for the space.

Findings

Structure of $(E_8)^{\sigma,\sigma'}$ determined

Space $E_8/(E_8)^{\sigma,\sigma'}$ identified as EVIII-VIII-VIII

Explicit involutions $\sigma, \sigma'$ provided

Abstract

For the simply connected compact exceptional Lie group $E_8$, we determine the structure of subgroup $(E_8)^{\sigma, \sigma'}$ of $E_8$ which is the intersection $(E_8)^\sigma \cap (E_8)^{\sigma'}$. Then the space $E_8/(E_8)^{\sigma, \sigma'}$ is the exceptional $\mathbb{Z}_2 \times \mathbb{Z}_2$- symmetric space of type EVIII-VIII-VIII, and that we give two involutions $\sigma, \sigma'$ for the space $E_8/(E_8)^{\sigma, \sigma'}$ concretely.