Given a subset $S\subseteq\{1,\cdots,n\}$ there is an associated flag manifold $F(S)$. Whenever $A\subseteq B$ there is a "forgetful" projection $F(A)\leftarrow F(B)$ (in fact I think its fibers are direct products of flag manifolds). Due to the following examples, I am curious when these projectios have sections:
- From complex vector spaces, we get sections $F(1,2n)\to F(1,2,2n)$.
- From quaternionic vector spaces, we get $F(1,4n)\to F(1,2,3,4,4n)$.
- More generally, I believe we get $F(1,n)\to F(1,2,\cdots,\rho(n),n)$ for spin representations, where $\rho(n)$ is the max number of linearly independent vector fields on a sphere $S^{n-1}$.
- From the exotic seven-dimensional binary and eight-dimensional ternary cross products we get the two sections $F(2,7)\to F(2,3,7)$ and $F(3,8)\to F(3,4,8)$.
The last example is suggestive: looking at coordinate axes, the sections induce the two Steiner triple systems $S(2,3,7)$ and $S(3,4,8)$. The other sections of $F(1,2n)\to F(1,2,2n)$ and $F(1,4n)\to F(1,k,4n)$ (for any of $k\in\{2,3,4\}$) induce boring Steiner triples.
Q1. Is every Steiner triple induced from a section of flag manifolds in this manner?
Q2. Does every section of flag manifolds have a "discrete" version on coordinate axes?
Q3. Are there any general results on when these sections exist and what shape they take?
Bonus. Is anything different if we consider oriented flag manifolds?
For (Q1), perhaps the first thing to consider would be the triple $S(2,3,9)$, the collection of affine lines in $\mathbb{F}_3^2$. Maybe it can be "linearized" via a map $\mathbb{R}X\otimes \mathbb{R}X\to \mathbb{R}X$ (where $X=\mathbb{F}_3^2$) much like the case of cross products. For (Q2), a priori it doesn't seem like we know there exists coordinates which the section plays nice with (i.e. sends coordinate subspaces to coordinate subspaces).
