The Grassmannian

The first thing we need to do to simplify our life is to get out of projective space. Recall that $\newcommand{\PP}{\mathbb{P}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\ZZ}{\mathbb{Z}}
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\Fl}{Fl}
\DeclareMathOperator{\GL}{GL}\PP^m$ can be defined as the collection of lines through the origin in $\CC^{m+1}$. Furthermore, lines in $\PP^m$ correspond to planes through the origin in $\CC^{m+1}$, and so on.

In problem $3$ in the introduction, we are trying to find lines in $\PP^3$ with certain intersection properties. This translates to a problem about planes through the origin in $\CC^4$, which we refer simply as $2$-dimensional subspaces of $\CC^4$. We wish to know which $2$-dimensional subspaces $V$ intersect each of four given $2$-dimensional subspaces $W_1,W_2, W_3, W_4$ in at least a line. Our strategy will be to consider the algebraic varieties $Z_i$, $i=1,\ldots,4$, of all possible $V$ intersecting $W_i$ in at least a line, and find the intersection $Z_1\cap Z_2\cap Z_3\cap Z_4$. Each $Z_i$ is an example of a Schubert variety, a moduli space of subspaces of $\CC^m$ with specified intersection properties.

The simplest example of a Schubert variety, where we have no constraints on the subspaces, is the Grassmannian
$
\Gr^n(\CC^m)$.

We will see later that the Grassmannian has the structure of an algebraic variety, and has two natural topologies that come in handy. For this reason we will call its elements the points of the $\Gr^n(\CC^m)$, even though they’re “actually” subspaces of $\CC^m$ of dimension $r=m-n$. It’s the same misfortune that causes us to refer to a line through the origin as a “point in projective space.”

Now, every point of the Grassmannian is the span of $r$ independent row vectors of length $m$, which we can arrange in an $r\times m$ matrix. For instance, the following represents a point in $\Gr^3(\CC^7)$.
$$\left[\begin{array}{ccccccc}
0 & -1 & -3 & -1 & 6 & -4 & 5 \\
0 & 1 & 3 & 2 & -7 & 6 & -5 \\
0 & 0 & 0 & 2 & -2 & 4 & -2
\end{array}\right]$$
Notice that we can perform elementary row operations on the matrix without changing the point of the Grassmannian it represents. Therefore:

Each point of the Grassmannian corresponds to a unique full-rank matrix in reduced row echelon form.

Let’s use the convention that the pivots will be in order from left to right and bottom to top.

Example. In the matrix above we can switch the second and third rows, and then add the third row to the first to get:
$$\left[\begin{array}{ccccccc}
0 & 0 & 0 & 1 & -1 & -2 & 0 \\
0 & 0 & 0 & 2 & -2 & 4 & -2 \\
0 & 1 & 3 & 2 & -7 & 6 & -5 \\
\end{array}\right]$$
Here, the bottom left $1$ was used as the pivot to clear its column. We can now use the $2$ at the left of the middle row as our new pivot, by dividing that row by $2$ first, and adding or subtracting from the two other rows:
$$\left[\begin{array}{ccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & -1 & 2 & -1 \\
0 & 1 & 3 & 0 & -5 & 2 & -3 \\
\end{array}\right]$$
Finally we can use the $1$ in the upper right corner to clear its column:
$$\left[\begin{array}{ccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & -1 & 2 & 0 \\
0 & 1 & 3 & 0 & -5 & 2 & 0 \\
\end{array}\right],$$
and we are done.

In the preceding example, we were left with a reduced row echelon matrix in the form
$$\left[\begin{array}{ccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & \ast & \ast & 0 \\
0 & 1 & \ast & 0 & \ast & \ast & 0 \\
\end{array}\right],$$
i.e. its leftmost $1$’s are in columns $2$, $4$, and $7$. The subset of the Grassmannian whose points have this particular form constitutes a Schubert cell.

Schubert varieties and cell complex structure

To make the previous discussion rigorous, we assign to the matrices of the form
$$\left[\begin{array}{ccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & \ast & \ast & 0 \\
0 & 1 & \ast & 0 & \ast & \ast & 0 \\
\end{array}\right]$$
a partition – a nonincreasing sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_r)$ – as follows. Cut out the “upside-down staircase” from the left of the matrix, and let $\lambda_i$ be the distance from the end of the staircase to the $1$ in each row. In the matrix above, we get the partition $\lambda=(4,2,1)$. Notice that we always have $\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_r$.

By identifying the partition with its Young diagram, we can alternatively define $\lambda$ as the complement in a $r\times n$ box (recall $n=m-r$) of the diagram $\mu$ defined by the $\ast$’s, where we place the $\ast$’s in the lower right corner. For instance:

Notice that every partition $\lambda$ we obtain in this manner must fit in the $r\times n$ box. For this reason, we will call it the Important Box. (Warning: this terminology is not standard.)

Notice that since each $\ast$ can be filled with any complex number, we have $\Omega_{\lambda}^\circ\cong \CC^{r\cdot n-|\lambda|}$. Thus we can think of the Schubert cells as forming an open cover of the Grassmannian by affine subsets.

More rigorously, the Grassmannian can be viewed as a projective variety by embedding $\Gr^n(\CC^m)$ in $\PP^{\binom{m}{r}-1}$ via the Plücker embedding. To do so, order the $r$-element subsets $S$ of $\{1,2,\ldots,m\}$ arbitrarily and use this ordering to label the homogeneous coordinates $x_S$ of $\PP^{\binom{m}{r}-1}$. Now, given a point in the Grassmannian represented by a matrix $M$, let $x_S$ be the determinant of the $r\times r$ submatrix determined by the columns in the subset $S$. This determines a point in projective space since row operations can only change the coordinates up to a constant factor, and the coordinates cannot all be zero since the matrix has rank $r$.

One can show that the image is an algebraic subvariety of $\PP^{\binom{m}{r}-1}$, cut out by homogeneous quadratic relations known as the Plücker relations. (See Miller and Sturmfels, chapter 14.) The Schubert cells form an open affine cover.

We are now in a position to define the Schubert varieties as closed subvarieties of the Grassmannian.

In general, however, we can use a different basis than the standard basis $e_1,\cdots,e_m$ for $\CC^m$. Given a complete flag, i.e. a chain of subspaces $$0=F_0\subset F_1\subset\cdots \subset F_m=\CC^m$$ where each $F_i$ has dimension $i$, we can define
$$\Omega_{\lambda}(F_\bullet)=\{V\in \mathrm{Gr}^n(\CC^m)\mid \dim V\cap F_{n+i-\lambda_i}\ge i.\}$$

Remark. The numbers $n+i-\lambda_i$ are the positions of the $1$’s in the matrix starting from the right. Combinatorially, without drawing the matrix, these numbers can be obtained by adjoining an upright staircase to the end of the $r\times n$ Important Box that $\lambda$ is contained in, and computing the distances from the right boundary of $\lambda$ to the right boundary of the enlarged figure.

Example. The Schubert variety $\Omega_{\square}(F_\bullet)\subset \Gr^{2}(\CC^4)$ is the collection of $2$-dimensional subspaces $V\subset \CC^4$ for which $\dim V\cap F_2\ge 1$, i.e. $V$ intersects another $2$-dimensional subspace (namely $F_2$)in at least a line.

By choosing four different flags $F^{(1)}_{\bullet},F^{(2)}_{\bullet},F^{(3)}_{\bullet},F^{(4)}_{\bullet}$, problem 3 becomes equivalent to finding the intersection of the Schubert varieties $$\Omega_{\square}(F^{(1)}_\bullet)\cap \Omega_{\square}(F^{(2)}_\bullet)\cap \Omega_{\square}(F^{(3)}_\bullet)\cap \Omega_{\square}(F^{(4)}_\bullet).$$

The CW complex structure

The Schubert varieties also give a CW complex structure on the Grassmannian for each complete flag as follows. Given a fixed flag, define the $0$-skeleton $X_0$ to be the $0$-dimensional Schubert variety $\Omega_{(n^r)}$. Define $X_2$ to be $X_0$ along with the $2$-cell (since we are working over $\CC$ and not $\RR$) formed by removing a corner square from the rectangular partition $(n^r)$, and the attaching map given by the closure in the Zariski topology on $\Gr^n(\CC^m)$. Continue in this manner to define the entire cell structure, $X_0\subset X_2\subset\cdots \subset X_{2nr}$.

This gives the second topology on the Grassmannian, and the one which is easier to work with in computing its cohomology.