Centralizers of the complex orthogonal and symplectic group

TL;DR

This paper presents a recursive algorithm to compute the centralizers of complex orthogonal and symplectic groups, revealing their structure as block Toeplitz matrices and enabling analysis of isotropy groups.

Contribution

It introduces a novel recursive method for determining centralizers and isotropy groups of these classical groups with respect to similarity transformations.

Findings

The algorithm precisely computes the centralizers of the groups.

Centralizers are shown to be conjugate to block Toeplitz matrix groups.

The method applies to spaces of skew-symmetric and Hamiltonian matrices.

Abstract

We find a recursive algorithm for computing the precise centralizers of the complex orthogonal and symplectic groups, and hence the isotropy groups, with respect to the similarity transformation on the spaces of skew-symmetric and Hamiltonian matrices, respectively. These groups are conjugate to groups of certain nonsingular block matrices whose blocks are rectangular block Toeplitz.