Schubert Cells of Flags

Question

I have been reading on these notes Undergraduate Lectures on Flag Varieties and I need some explanations on two things:

1. In page 3, how he modefied the matrices in the "Second Attempt"

2. In the same manner, how can I get the flag $Fl(1,2,3)(\mathbb C^3)$?

Thanks for your help!

Answer 1

For a quick and dirty explanantion that offers no insight (as requested), take a $3\times 3$ permutation matrix and draw a star in every unoccupied spot that is not above (in the same column) or to the right (in the same row) of any $1$. Put $0$s everywhere else. That's the Schubert cell corresponding to the permutation matrix.

For example, for the permutation 312 (which has length $2$, so the cell should have codimension $2$, dimension $1$) we have $$\left[\begin{array}{ccc}0&0&1\\1&0&0\\\ast&1&0\end{array}\right]$$ and for 213 (length $1$, codimension $1$, dimension $2$) we have $$\left[\begin{array}{ccc}0&1&0\\1&0&0\\\ast&\ast&1\end{array}\right]$$