Special orthogonal, special unitary, and symplectic groups as products of Grassmannians

TL;DR

This paper reveals that special orthogonal, special unitary, and symplectic groups can be represented as products of their respective Grassmannians, uncovering a novel structural relationship among these classical groups.

Contribution

It introduces a new way to express these groups as products of Grassmannians realized as involution matrices, a previously unobserved structural property.

Findings

SO(n) is a product of two real Grassmannians

SU(n) is a product of four complex Grassmannians

Sp(2n, R) and Sp(2n, C) are products of four symplectic Grassmannians

Abstract

We describe a curious structure of the special orthogonal, special unitary, and symplectic groups that has not been observed, namely, they can be expressed as matrix products of their corresponding Grassmannians realized as involution matrices. We will show that $\operatorname{SO}(n)$ is a product of two real Grassmannians, $\operatorname{SU}(n)$ a product of four complex Grassmannians, and $\operatorname{Sp}(2n, \mathbb{R})$ or $\operatorname{Sp}(2n, \mathbb{C})$ a product of four symplectic Grassmannians over $\mathbb{R}$ or $\mathbb{C}$ respectively.