Decomposition of spinor groups by the involution {\sigma}' in exceptional Lie groups

TL;DR

This paper determines the structure of fixed subgroups of spinor groups under a specific involution within exceptional Lie groups, enhancing understanding of their subgroup decompositions.

Contribution

It explicitly identifies the group structures of fixed subgroups under involution {\sigma'} in various spinor groups within exceptional Lie groups.

Findings

Fixed subgroup structures of Spin(n) under {\sigma'} are explicitly determined.

The involution {\sigma'} induces specific subgroup decompositions.

Results contribute to the understanding of symmetry and subgroup structure in exceptional Lie groups.

Abstract

The compact exceptional Lie groups F4, E6, E7 and E8 have spinor groups as a subgroup as follows: E8 \supset Ss(16) \supset Spin(15) \supset Spin(14) \supset Spin(13), E7 \supset Spin(12) \supset Spin(11), E6 \supset Spin(10), F4 \supset Spin(9) \supset Spin(8) \supset Spin(7) \supset \cdot \cdot \cdot \supset Spin(1) \ni 1. We know the involution {\sigma}' induced an element {\sigma}' \in Spin(8) \subset F4 \subset E6 \subset E7 \subset E8. Now, in this paper, we determine the group structures of (Spin(n)){\sigma}' which are the fixed subgroups by the involution {\sigma}'.