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4.ii.a The Geometry of Grassmann Varieties

We develop further geometric properties of Grassmann varieties. The kth exterior power of the embedding K --> Cn of a k-plane K into Cn gives the embedding


whose image is a 1-dimensional subspace of the kth exterior power of Cn, and hence a point in the projective space PM-1, where M is the number of k subsets of the numbers 1, 2, ..., n, the binomial coefficient n!/k!(n-k)!. This point determines the k-plane K uniquely. The Plücker embedding is the resulting projective embedding of the Grassmannian

Gr(k, n)   ---->   PM-1.

The n!/k!(n-k)! homogeneous Plücker coordinates for the Grassmannian in this embedding are realized concretely as follows. Represent a k-plane K as the row space of a k by n matrix, also written K. A maximal minor of K = (xij) is the determinant of a k by k submatrix of K: Given a choice of columns a : a1 < a2 < ... < ak with ak at most n, set pa(K) := det (K|a), where K|a is the submatrix of K consisting of the columns from a:

The vector (pa(K)) of N maximal minors of K defines the map (4.4) giving Plücker coordinates for K. Let Cn,k be the collection of these indices of Plücker coordinates.

The indices a, b in Cn,k have a natural Bruhat order

b \leq a   <==>   bj \leq aj for every j from 1 to k .

The Schubert variety Xa is
Xa   =   {K in Gr(k, n) | pb(K) = 0 whenever b is not less than or equal to a} . (4.5)

This has dimension |a| := a1-1 + a2-2 + ... + ak-k.

The relevance of the Plücker embedding to Question 4.3 when each li equals 1 is seen as follows. Let L be a (n-k)-plane, represented as the row space of a (n-k) by n matrix, also written L. Then a general k-plane K meets L non-trivially if and only if


Laplace expansion along the rows of K gives
    =   pa(K) La ,

where the sum is over all a in Cn,k and La is the appropriately signed minor of L given by the columns complementary to a. Hence the set X(L) of k-planes that meet the (n-k)-plane L non-trivially is a hyperplane section of the Grassmannian in its Plücker embedding.

Thus the set of k-planes meeting k(n-k) general (n-k)-planes non-trivially is a complementary linear section of the Grassmannian, and so the number d(k,n) of such k-planes is the degree of the Grassmannian in its Plücker embedding. More generally, if a in Cn,k and L1, L2, ..., L|a| are general (n-k)-planes, then the number of points in the intersection of the Schubert varieties

X(L1), X(L2), ..., X(L|a|) (4.7)

is the degree d(a) of the Schubert variety Xa, which we compute in the next section.
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Next: 4.ii.b. Degrees of Schubert Varieties
Up: 4.ii The Special Schubert Calculus