## Schubert Varieties

It’s been awhile since the last post, but Spring Break happened.  Anyway, back to Schubert Calculus! Last time, we discussed Grassmannians, this time, we’re going to talk about their most important subvarieties, the Schubert Varieties.

We’ll start by taking $F_*$ to be a complete flag.  That is, it’s an increasing sequence $0\subset F_1\subset F_2\subset\ldots\subset F_m=E$ of subspaces of a vector space $math E$ where $\dim F_i=i$.  We’re going to work on $Gr^n(E)$, and so we have $r=m-n$, and the Grassmannian parameterizes the codimension $n$ subspaces, which is equivalent to parameterizing the $r$-dimensional subspaces of $E$.

Now, we take a strictly decreasing sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_r)$ with $\lambda_i\leq n$.  We want to picture this as a subset of an $r\times n$ grid, with $\lambda_i$ boxes in the $i$th row.  For example, if we had the sequence $\lambda=(5,4,1,0,\ldots,0)$, we would have the diagram below. Young Diagram (5,4,1)

This way of looking at things will be helpful later when we’re trying to do computations.  These are called Young Diagrams.

So now, given the flag $F_*$ and the Young Diagram $\lambda$, we can define a subvariety of $Gr^n(E)$ as follows: $\Omega_\lambda(F_*)=\{V\in Gr^n(E)|\dim(V\cap F_{n+i-\lambda_i}\geq i,1\leq i\leq r\}$.

These are called Schubert varieties, and they depend on both the flag and the Young Diagram.  So first up, some basic facts.  The reason that the Young Diagram has to fit inside an $r\times n$ grid is because $|\lambda|=\sum_i \lambda_i$ is the codimension of the Schubert Variety, and you can’t have codimension greater than the dimension of $Gr^n(E)$, which is $rn$.  Also, they’re irreducible closed subvarieties.  We’ll prove this:

Define the Schubert Cell $\Omega_\lambda^\circ$ to be the locus in $Gr^n(E)$ satisfying $\dim(V\cap F_k)=i$ for $n+i-\lambda_i\leq k\leq n+i-\lambda_{i+1}$ for $0\leq i\leq r$.  To see what this gets us, we take $e_1,\ldots,e_m$ a basis for $E$ and take $F_i=\mathrm{span}(e_1,\ldots, e_i)$.  Then and $V\in\Omega_\lambda^\circ$ is spanned by the rows of a unique $r\times m$ matrix in the reduced row eschelon form with a $1$ in the $n+i-\lambda_i$ position from the left in row $i$.

An example, taken from Fulton’s “Young Tableaux,” is that if we take $r=5$ and $n=7$, with $\lambda(5,3,2,2,1)$, we’ll get the matrices of the form $\left(\begin{array}{cccccccccccc}*&*&1&0&0&0&0&0&0&0&0&0\\ *&*&0&*&*&1&0&0&0&0&0&0\\ *&*&0&*&*&0&*&1&0&0&0&0\\ *&*&0&*&*&0&*&0&1&0&0&0\\ *&*&0&*&*&0&*&0&0&*&1&0\end{array}\right)$

This means that $\Omega^\circ_\lambda$ is an affine space, and it’s dimension is $rn-|\lambda|$, and as the notation suggests, $\Omega_\lambda^\circ\subset\Omega_\lambda$ is an open dense subset, and so it determines the dimension, and the closure of an irreducible space is irreducible.

That’ll be all for right now, the real stuff is what’s coming up next: getting cohomology classes out of the Schubert varieties, then learning the Pieri and Giambelli formulas which compute the ring structure on $H^*(Gr^n(E),\mathbb{Z})$, and finally, using it to solve problems involving the enumeration of lines (mostly, it works in higher dimensions, but gets computationally unpleasant). Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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### 5 Responses to Schubert Varieties

1. cornelioid says:

Reading pleasantly! I gather we should take $\lambda_{0}$ to be $n$. Furthermore, while i don’t have a copy of Fulton, the dimension conditions established on the $\Omega^{\circ}_{\lambda}$ in the fifth paragraph seem to give the cell dimension $|\lambda|$; should the condition not instead be $i+\lambda_{r-i}\leq k\leq i+\lambda_{r-i-1}$? Accordingly, the matrix seems reversed, with $F_{i}$ confined to the rightmost $i$ columns. (Pardon me if i’m just not seeing straight.)

Some minor typos: Second paragraph, name of vector space missing “latex” within $s. Fourth paragraph, math insert, missing right paren in dimension. Fifth paragraph, stray “and” following “Then”. 2. Charles Siegel says: Yeah, $\lambda_0=n$ and $\lambda_{r+1}=0$. As for what you wrote, I think you’re incorrect here. With the conditions I gave, in that example, I get that the space must have no intersection with $\langle e_1,e_2\rangle$, must have dimension 1 with$langle e_1,\ldots,e_5\rangle\$, etc. So it’s the span of some set of five row vectors which can be put into the form in the matrix above. That matrix has 22 free parameters, and so says that the cell has dimension 22. The weight of the partition is 13=5+3+2+2+1, and the sum, 22+13=35=7*5, the total dimension of the Grassmannian $Gr^5(\mathbb{C}^{12})$. Which means that this Schubert cell has codimension $13=|\lambda|$, as I said.

3. cornelioid says:

You’re right! Sorry about that. This same point tripped me up when i first read Fulton.