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We describe matrix products of Grassmannians for special orthogonal, unitary, and symplectic groups, indicating a novel structure.
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.
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Lim et al. (2026) studied this question.
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