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Schubert Calculus II: Schubert Stratification

It’s time to set up the Schubert stratification of $G(k,n)$, a decomposition of the whole variety into locally closed pieces (Schubert cells), each of which is isomorphic to an affine space.

These cells are indexed by partitions and, by general theory, give rise to an additive basis of the Chow, cohomology, and $K$-theory rings of the Grassmannian. And, as a result, intersection theory on $G(k,n)$ has a complete, concrete description in terms of the combinatorics of the Schubert decomposition (i.e. partitions and the Littlewood-Richardson rule).

Schubert Cells, Varieties and Classes

A Schubert cell on $G(k,n)$ is defined relative to a choice of complete flag $\mathcal{F} : F_1 \subset F_2 \subset \cdots \subset F_n = \mathbb{C}^n$ of vector subspaces, with $\dim F_i = i$, by specifying how much our $k$-plane should intersect each portion of the flag.

In particular, let $d = (d_1 \leq d_2 \leq \cdots \leq d_n)$ be a “valid sequence”: it should start at $0$ or $1$, grow by at most $1$ each step, and have $d_n = k$. Then the corresponding Schubert cell is

$\Omega_d^\circ = \{V \in G(k,n) : \dim (V \cap F_i) = d_i\}.$

This is a locally-closed condition, corresponding to certain determinants being zero and others being nonzero. The closure of the cell is the corresponding Schubert variety:

$\Omega_d^\circ(\mathcal{F}) = \{V \in G(k,n) : \dim (V \cap F_i) \geq d_i\}.$

This corresponds only to the vanishing of certain determinants. Finally, the cohomology class (or Chow class) of å Schubert variety is referred to as a Schubert class.

Observe that the Grassmannian is the disjoint union of all the Schubert cells, and that there are ${n \choose k}$ cells in all, since a valid dimension sequence contains exactly $k$ “jump points” where the dimension increases.

Important digression. the description in terms of dimension sequences, while intuitive to geometrically-minded people like me, is not the one in common use. There’s a good reason for this, which is that the numbers are unnecessarily complicated. For example, the dimension sequence for a generic $k$-plane is

$d = \underbrace{0, \ldots, 0}_{n-k},1,2,\ldots,k$,

and the dimension sequence for the “least generic” choice, namely $V = F_i$ itself, is

$d = 1, \ldots, k, \underbrace{k, \ldots, k}_{n-k}$.

So the numbers are growing as the $k$-plane specializes, but even the “simple” sequence still adds up to $\frac{k(k+1)}{2}$. There’s a better indexing system, which describes the dimensions by comparison with the generic case.

Specifically, for each $1 \leq i \leq k$, suppose that the integer $i$ shows up $\lambda_i$ steps earlier than in the generic case. Then the $\lambda_i$ are weakly decreasing, that is, they form a partition $\lambda$. We’ll index the cell by $\lambda$ instead.

By definition, the generic case now corresponds to the empty partition $(0,\ldots,0),$ which is much better! The special case corresponds to the rectangle partition $(n-k, \ldots, n-k).$ And in general, $\lambda$ is a partition that fits inside this $k \times(n-k)$ rectangle.

Under this system, the codimension of the cell corresponding to $\lambda$ is $|\lambda|$, which is good as well. We’ll use the partition indexing convention from now on.

The Structure of Schubert Cells

As stated above, Schubert cells are isomorphic to affine spaces. To see this, we’ll use the coordinate system worked out last post, with the “forward flag” on our basis $e_1, \ldots, e_n$ of $\mathbb{C}^n$:

$\mathcal{F} = \langle e_1 \rangle \subset \langle e_1, e_2 \rangle \subset \cdots \subset \langle e_1, \ldots, e_n \rangle$.

First consider the generic case, where $V$ intersects $\mathcal{F}$ as little as possible. Let us try to build a basis for $V$, a $k \times n$ row matrix.

Well, generically, $V \cap F_i = 0$ for $i = 1, \ldots, n-k$, but $V \cap F_{n-k+1} \ne 0$, so the first row of our matrix will be of the form

$\begin{bmatrix} \star & \star & \cdots & \star & 1 & 0 & \cdots & 0 \end{bmatrix}$,

The coefficient of $e_{n-k+1}$ has been scaled to $1$ (we know it’s nonzero, since there’s no vector purely in terms of the earlier basis elements) and none of the later basis elements show up.

Each of the subsequent rows will include one extra new basis element, giving us, in the end, a matrix like this (in $G(3,6)$):

$\begin{bmatrix} \star & \star & \star & 1 & 0 & 0 \\ \star & \star & \star & \star & 1 & 0 \\ \star & \star & \star & \star & \star & 1 \\ \end{bmatrix}$.

Finally, we use row operations to clear out the $\star$‘s below the ones:

$\begin{bmatrix} \star & \star & \star & 1 & 0 & 0 \\ \star & \star & \star & 0 & 1 & 0 \\ \star & \star & \star & 0 & 0 & 1 \\ \end{bmatrix}$.

This is now a canonical representative: every $V$ in the cell can be represented uniquely in this way, with the final columns forming an identity matrix. (Any additional row operations would change this identity matrix.) Incidentally, we see that the generic cell coincides with the standard affine chart (defined in the last post) corresponding to the nonvanishing of the final Plücker coordinate. It is isomorphic to $\mathbb{A}^{k(n-k)}$.

What about the other cells? Well, if $\lambda_i > 0$, then the one in the $i$-th row occurs that many spaces more to the left. For example, if $\lambda = (3,1)$, the resulting matrix looks like this:

$\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0\\ 0 & \star & \star & 1 & 0 & 0 \\ 0 & \star & \star & 0 & \star & 1 \\ \end{bmatrix}$,

so now the identity matrix is formed by the first, fourth and sixth columns. The pictured cell is isomorphic to $\mathbb{A}^5$, which is of codimension $|\lambda| = 4$ inside of $G(3,6)$. (In general, there are zeros to the right of the ones and in the same columns as the ones, and $\star$‘s everywhere else.)

One useful computation is the following: if we used a “backwards flag”,

$\mathcal{G} = \langle e_n \rangle \subset \langle e_{n-1}, e_n \rangle \subset \cdots \subset \langle e_1, \ldots, e_n \rangle$,

then the flags $\mathcal{F},\mathcal{G}$ intersect transversely, as do the corresponding Schubert varieties. (In the chart, we’re just setting various coordinates to 0.) The cells for $\mathcal{G}$ look just like those for $\mathcal{F}$, but reversed left-to-right and (to rearrange the ones into an identity matrix) the rows are also reversed in order:

$\begin{bmatrix} 1 & 0 & 0 & \star & \star & \star \\ 0 & 1 & 0 & \star & \star & \star \\ 0 & 0 & 1 & \star & \star & \star \end{bmatrix} \ \ \ \begin{bmatrix} 1 & \star & 0 & \star & \star & 0 \\ 0 & 0 & 1 & \star & \star & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix},$
Generic and $\lambda = (3,1)$ Schubert cells for $\mathcal{G}$

Note that $\lambda_1 = 3$ moved the bottom row’s $1$ to the right by $3$ steps. Overall, the matrix has zeros to the left of the ones and in the same columns as the ones, and $\star$‘s everywhere else.

The key thing to see is that for a suitable choice of $\lambda, \mu$, we can make the cells $\Omega_\lambda(\mathcal{F}), \Omega_\mu(\mathcal{G})$ intersect in a single point, by lining up the identity matrix columns. It turns out that this happens when $\lambda$ and $\mu$ are “complements” of each other inside the $k \times (n-k)$ rectangle. (One should be rotated around 180 degrees.)

In this case the corresponding Schubert varieties intersect transversely in a single point, which will have important consequences in cohomology.

Cohomological Consequences

So this is the Schubert stratification. It has an important organizational property (which makes it worthy of being called a “stratification”): the closure of each cell is a union of other cells. (In particular, the closure of $\Omega_\lambda^\circ$ is $\bigcup_{\mu \subseteq \lambda} \Omega_\mu^\circ$.) Put another way, if the closure of cell $A$ touches cell $B$, then in fact it contains $B$.

This is a so-called affine stratification, and it has big consequences for the cohomology of the space:

Theorem. Let $X$ be a smooth projective variety, with an affine stratification into cells $X_i^\circ$ isomorphic to affine spaces. Let $X_i = \overline{X_i^\circ}$ be the closure of the cell. Then:
(1) The classes $[X_i]$ form an additive basis for the cohomology ring $H^*(X)$.
(2) The classes $[X_i]$ form an additive basis for the Chow ring $A(X)$. In particular, the natural map $A(X) \to H^*(X)$ is an isomorphism.
(3) The classes of the structure sheaves $[\mathcal{O}_{X_i}]$ form an additive basis for the $K$-theory ring $K(X)$.

The idea of this proof is that $\mathbb{A}^n$ is contractible, both topologically and algebraically. So, if $Y$ is properly contained in $\mathbb{A}^n$, we can use a homotopy to push off to infinity. Then it collides with the boundary, which by the stratification property is a union of smaller cells. Either it fills up the cells (in which case it is cohomologous to them) or is properly contained in them — we repeat until done. (For K-theory, adapt this argument to sheaves instead of cycles.)

Additionally, in our case, the computation that showed $\Omega_\lambda(\mathcal{F}) \cap \Omega_\mu(\mathcal{G}) = \{\text{pt}\}$ means that the Schubert basis is self-dual under the intersection product of the cohomology ring. We’ll make use of this property when we do some enumerative geometry in the next post.

That’s all for now. Coming up next: computations in cohomology and enumerative geometry!