Schubert cell decomposition and full flags

Question

I am looking for a self-contained basic theory of Schubert-cells through finding the decomposition of the full flag $Fl_3(\mathbb C^3)$.

Answer 1

Usually Schubert cells are constructed using the structure theory of semisimple Lie groups (in this case $SL(3,\mathbb C)$), but for this case, one can give an explicit description as follows. Start by fixing one flag, say the standard one $\mathbb C\subset\mathbb C^2\subset\mathbb C^3$. This will be the unique Schubert-cell of dimension $0$. The further cells can be indexed by the dimensions of the intersections of their constituents with the spaces in the standard flag.

Explicitly, there are two Schubert-cells of dimension $1$. One of those consists of flags $\ell\subset\mathbb C^2$ with $\ell\neq\Bbb C$. This is the complement of the point defined by $\mathbb C$ in $\mathbb CP^2$ and hence diffeomorphic to $\mathbb C$. The second cell of dimension one consists of flags of the form $\mathbb C\subset W$ with $W\neq\mathbb C^2$. This again looks like the complement of a point in the projectivization of $\mathbb C^3/\mathbb C$, and hence like $\mathbb C$.

From now on, we only have to look at flags $\ell\subset W$, where $\ell\neq\mathbb C$ and $W\neq\mathbb C^2$. There are two Schubert-cells of Dimension $2$. One of them is formed by those flags, for which $\mathbb C\subset W$. Since $\ell\neq\mathbb C$ and $W\neq\mathbb C^2$, we get $W=\mathbb C\oplus\ell$. Hence the flag is determined by the line $\ell$, which is not contained in $\mathbb C^2$. So this looks like $\mathbb CP^3\setminus\mathbb CP^2\cong\mathbb C^2$. The second of these cells consists of those flags, for which $\ell\subset\mathbb C^2$. Since $W\neq\mathbb C^2$, this implies that $\ell=W\cap\mathbb C^2$, so this time, the flag is determined by the plane $W$, which should not contain $\mathbb C$. Again this shows that the space of such planes is isomorphic to $\mathbb C^2$.

Finally, there is one cell of dimension $3$ consisting of the "generic" flags, for which $\ell$ is not contained in $\mathbb C^2$, and $\mathbb C$ is not contained in $W$. Such a flag is determined by $\ell$, which lies in $\mathbb CP^3\setminus\mathbb CP^2$ and $W\cap\mathbb C^2$, which lies in the complement of a point in $\mathbb CP^2$. Hence the space of generic flags looks like $\mathbb C^3$.