What question did this study set out to answer?

June 26, 2026Open Access

Special Orthogonal, Special Unitary, and Symplectic Groups as Products of Grassmannians

Key Points

To unveil the unique structure of special orthogonal, special unitary, and symplectic groups expressed as products of Grassmannians.
Characterized special orthogonal, unitary, and symplectic groups using their corresponding Grassmannians.
Realized these Grassmannians as involution matrices for the respective groups.
Demonstrated the product structures for each type of group.
Identified that SO(n) can be expressed as a product of two real Grassmannians.
Demonstrated that SU(n) can be represented as a product of four complex Grassmannians.
Established that Sp(2n, R) or Sp(2n, C) can be described as a product of four symplectic Grassmannians over R or C.

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 SO (n) is a product of two real Grassmannians, SU (n) a product of four complex Grassmannians, and Sp (2n, R) or Sp (2n, C) a product of four symplectic Grassmannians over R or C respectively.

Ask AI

Helpful

Bookmark

View Full Paper