3
$\begingroup$

I have a question about the the proof of Pieri's formula from Harris' and Eisenbuds's lecture "3264 and All That"on page 146. Before the proof we use this terminology (see page 139):

let $G=G(k,V)$ be the Grassmannian of $k$-dimensional linear subspaces of an $n$-dimensional vector space $V$ . We start with one useful definition. As we said, Kleiman’s theorem assures us (in characteristic $0$) that, for a general pair of flags $\mathcal{V}= 0=V_0 \subset V_1 \subset ... \subset V_n=V$ and $\mathcal{W}$ on $V$ , the Schubert cycles $\Sigma_a(\mathcal{V}),\Sigma_b(\mathcal{W})$ intersect generically transversely.

Recall, these Schubert cycles are defined as follows:

Let $n-k \ge a_1 \ge a_2 \ge ... \ge a_k \ge 0$ a descending sequence considered as vector $a=(a_1,...,a_k)$. Then

$$\Sigma_a(\mathcal{V}):= \{\Lambda \in G \vert \dim(\Lambda \cap V_{n-k+i-a_i}) \ge i \text{ for all } i \}$$

In this case, we can actually say explicitly what “general” means:

Definition 4.4. We say that a pair of flags $\mathcal{V}$ and $\mathcal{V}$ on $V$ are transverse if any of the following equivalent conditions hold:

(a) $V_i \cap W_{n-i} =0$ for all $i$ .

(b) $\dim(V_i \cap W_{n-i})= \min(0, i +j-n)$ for all $i, j$ .

(c) There exists a basis $e_1,...,e_n$ for $V$ in terms of which

$$V_i =\langle e_1,...,e_i \rangle \text{ and } W_j =\langle e_{n+1-j},...,e_n \rangle$$

Additionally, we will use:

Lemma 4.5 Let $\Sigma_a (\mathcal{V}), \Sigma_b (\mathcal{W}) \subset G$ be Schubert cycles defined relative to transverse flags V and W on V . If $\Lambda \in \Sigma_a (\mathcal{V}) \cap \Sigma_b (\mathcal{W})$ is a general point of their intersection, then:

(a) $\Lambda$ does not lie in any strictly smaller Schubert cycle $ \Sigma_{a'} (\mathcal{V}) \subset \Sigma_a (\mathcal{V})$.

(b) The flags induced by $\mathcal{V}$ and $\mathcal{W}$ on $\Lambda$ (that is, consisting of intersections with $\Lambda$ with flag elements $V_{\alpha}$ and $ W_{\beta}$ ) are transverse. Note that, by the first part, the flags $\Lambda^{\mathcal{V}}$ and $\Lambda^{\mathcal{W}}$ on $\Lambda$ induced by $\mathcal{V}$ and $\mathcal{V}$ are, explicitly,

$$\Lambda^{\mathcal{V}}_i= \Lambda \cap V_{n-k+i-a_i}$$

and

$$\Lambda^{\mathcal{W}}_i= \Lambda \cap V_{n-k+i-b_i}$$

for $i=1,...,k$.

now

Proposition 4.9 (Pieri’s formula). For any Schubert class $\sigma_a \in A(G) $(the latter is the Chow Group) and any integer $b$,

$$(\sigma_b \cdot \sigma_a)= \sum_{\vert c \vert= \vert a \vert + b \text{ & } a_i \le c_i \le a_{i-1}}$$

Proof of 4.9(Pieri's formula): [...]

following the book we look at Schubert cycles $\Sigma_a (\mathcal{V}), \Sigma_b (\mathcal{U})$ and $\Sigma_{c^*}(\mathcal{W})$ defined with respect general flags $\mathcal{V},\mathcal{U},\mathcal{W}$. Recall that $c^*=(c^*_1,c^*_2,..., c^*_k$ is the dual vector to $c$ defined by $c^*_i:= n-k -c_{k+1-i}$.

By definition,

$$\Sigma_a(\mathcal{V})= \{\Lambda \vert \dim(\Lambda \cap V_{n-k+i-a_i}) \ge i \text{ for all } i \}$$

and

$$\Sigma_{c^*}(\mathcal{W})= \{\Lambda \vert \dim(\Lambda \cap W_{i+c_{k+1-i}}) \ge i \text{ for all } i \}$$

Set

$$A_i = V_{n-k+i-a_i} \cap W_{k+1-i-c_i}$$.

During the proof we define $A:= \langle A_1,...,A_k \rangle$ and it is claimed that by Lemma 4.5 the plane $\Lambda$ is spanned by it's intersections with $A_i$, i.e. that $\Lambda =\langle \Lambda \cap A_i \vert i \rangle$.

Q: why Lemma 4.5 implies that the plane $\Lambda$ is spanned by it's intersections with $A_i$? do we really need the transversality of the flags for $\Lambda =\langle \Lambda \cap A_i \vert i \rangle$?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy