Orbit decomposition of Jordan matrix algebras of order three under the automorphism groups

TL;DR

This paper explicitly describes how the automorphism group acts on the real split Jordan algebra of 3x3 hermitian matrices, using the cross product and characteristic polynomial, to decompose the algebra into orbits.

Contribution

It provides an explicit orbit decomposition of the Jordan algebra under automorphisms, extending understanding of symmetry actions on these algebraic structures.

Findings

Explicit orbit decomposition formulas derived

Connection between automorphism groups and algebraic invariants clarified

Enhanced understanding of symmetry in Jordan algebras achieved

Abstract

The orbit decomposition is given under the automorphism group on the real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the automorphism group on the complexification, explicitly, in terms of the cross product of H. Freudenthal and the characteristic polynomial.