Let $l_1$ and $l_2$ be two lines in $G(1,n)$, the Grassmannian of lines in n dimensional projective space.
Suppose that their Plucker embeddings has dot product zero. Namely if $(x_1, x_2, \cdots, x_N)$
and $(y_1, y_2, \cdots, y_N)$ are their Plucker coordinates, $\sum_{i=1}^N x_i y_i = 0$.
Is there anything special about $l_1$ and $l_2$, e.g. are they necessarily orthogonal to each other?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
In a real vector space you can say the following. The lines you refer to correspond to planes $P$, $Q$ in $\mathbb{R}^{n+1}$. The dot product of the Plucker coordinates is zero if and only if $P$ intersects the orthogonal complement of $Q$ nontrivially. This doesn't mean the lines are orthogonal. In fact they need not intersect, e.g. if $P$ is contained in the orthogonal complement to $Q$.
-
$\begingroup$ No reason to specialize to $\mathbb{R}$. This is right over any field (maybe characteristic not $2$) using the standard inner product $\sum u_i v_i$. $\endgroup$ Commented May 29, 2013 at 15:46