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 to be a *complete flag*. That is, it’s an increasing sequence of subspaces of a vector space $math E$ where . We’re going to work on , and so we have , and the Grassmannian parameterizes the codimension subspaces, which is equivalent to parameterizing the -dimensional subspaces of .

Now, we take a strictly decreasing sequence of nonnegative integers with . We want to picture this as a subset of an grid, with boxes in the th row. For example, if we had the sequence , 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 and the Young Diagram , we can define a subvariety of as follows:

.

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 grid is because is the codimension of the Schubert Variety, and you can’t have codimension greater than the dimension of , which is . Also, they’re irreducible closed subvarieties. We’ll prove this:

Define the Schubert Cell to be the locus in satisfying for for . To see what this gets us, we take a basis for and take . Then and is spanned by the rows of a unique matrix in the reduced row eschelon form with a in the position from the left in row .

An example, taken from Fulton’s “Young Tableaux,” is that if we take and , with , we’ll get the matrices of the form

This means that is an affine space, and it’s dimension is , and as the notation suggests, 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 , and finally, using it to solve problems involving the enumeration of lines (mostly, it works in higher dimensions, but gets computationally unpleasant).

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About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.

Reading pleasantly! I gather we should take to be . Furthermore, while i don’t have a copy of Fulton, the dimension conditions established on the in the fifth paragraph seem to give the cell dimension ; should the condition not instead be ? Accordingly, the matrix seems reversed, with confined to the rightmost 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”.

Yeah, and .

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 , 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 . Which means that this Schubert cell has codimension , as I said.

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

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