Random matrices beyond the Cartan classification

TL;DR

This paper extends the classification of random matrix ensembles beyond Hermitian cases to include many non-Hermitian ensembles with physical symmetries, linking them to symmetric spaces and providing explicit representations.

Contribution

It demonstrates that numerous non-Hermitian random matrix ensembles with physical symmetries can be classified as symmetric spaces, expanding the Cartan classification.

Findings

Identified non-Hermitian ensembles with symmetric spaces

Explicit representations of ensembles and invariance groups

Connected Hermitian and Ginibre ensembles within the classification

Abstract

It is known that hermitean random matrix ensembles can be identified with symmetric coset spaces of Lie groups, or else with tangent spaces of the same. This results in a classification of random matrix ensembles as well as applications in practical calculations of physical observables. In this paper we show that a large number of non-hermitean random matrix ensembles defined by physically motivated symmetries - chiral symmetry, time reversal invariance, space rotation invariance, particle-hole symmetry, or different reality conditions - can likewise be identified with symmetric spaces. We give explicit representations of the random matrix ensembles identified with lateral algebra subspaces, and of the corresponding symmetric subalgebras spanning the group of invariance. Among the ensembles listed we identify as special cases all the hermitean ensembles identified with Cartan classes of symmetric spaces and the three Ginibre ensembles with complex eigenvalues.