Let $G(q,D)$ be the Grassmannian of $q$-dimensional vector spaces in $\mathbb{R}^D$, where $q \le D$ are positive integers. In the paper, a distance between two points of $G(q,D)$ are defined as follows. The authors represented a point in $G(q,D)$ by a matrix $X$ of size $D \times q$ such that $X^T X$ is a $q \times q$ identity matrix. For two points $X_1, X_2 \in G(q,D)$, they define $d(X_1,X_2) = 1/\sqrt{2} || X_1X_1^T - X_2X_2^T ||_F$, where $||\cdot ||_F$ is the matrix Frobenius norm. Could $d(X_1,X_2)$ be defined using Plücker coordinates in $\mathbb{C}[G(q,D)]$? Thank you very much.
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asked Jul 10, 2020 at 9:41
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$\begingroup$ Clearly this can be done when $q=1$, since the authors' coordinates parametrise lines in $\mathbb{R}^D$ by vectors of Euclidean norm $1$ (so the only choice is a sign) and the Plücker coordinates are just the usual projective coordinates, which can be normalized in the same way. $\endgroup$– Mark WildonCommented Jul 10, 2020 at 10:59
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