Special orthogonal, special unitary, and symplectic groups as products of Grassmannians
Lek-Heng Lim, Xiang Lu, Ke Ye
TL;DR
This work reveals a novel structure: fundamental classical groups can be expressed as products of Grassmannians realized as involution matrices. By developing canonical forms and dimension analyses for products of real, complex, and symplectic Grassmannians, the authors prove $\operatorname{SO}(n)=\Phi(\lfloor n/2\rfloor,\lfloor n/2\rfloor,\mathbb{R}^n)$, $\operatorname{SU}(n)=\Phi(\lfloor n/2\rfloor,\lfloor n/2\rfloor,\lfloor n/2\rfloor,\lfloor n/2\rfloor,\mathbb{C}^n)$, and $\operatorname{Sp}(2n,\mathbb{F})=\Phi_{\mathrm{Sp}}(2\lfloor n/2\rfloor,2\lfloor n/2\rfloor,2\lfloor n/2\rfloor,2\lfloor n/2\rfloor,\mathbb{F}^{2n})$, with precise dimension counts and optimality statements. They provide constructive algorithms for the two- and four-factor decompositions in the real and complex settings, and establish a rigorous bridge between Grassmannian products and symplectic Grassmannians, including a homogeneous-space interpretation. The paper also discusses model-independence and Lie-theoretic analogies, and raises open questions about broader extensions, including quaternionic Grassmannians. Overall, the results uncover unexpected yet structured decompositions of classical groups, with potential computational and theoretical implications in geometry and representation theory.
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.