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 ith 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)

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).

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