We develop further geometric properties of Grassmann varieties.
The kth exterior power of the embedding
K > C^{n} of a kplane K into
C^{n} gives the embedding
(4.4) 
The n!/k!(nk)! homogeneous Plücker
coordinates for the Grassmannian in this embedding are realized concretely as
follows.
Represent a kplane K as the row space of a
k by n matrix, also written K.
A maximal minor of K = (x_{ij})
is the determinant of a k by k submatrix
of K:
Given a choice of columns
a : a_{1} < a_{2} < ... <
a_{k} with a_{k} at most n, set
p_{a}(K) := det (K_{a}),
where K_{a} is the submatrix of K consisting of
the columns from a:
The indices a, b in
C_{n,k} have a natural Bruhat order
X_{a} = {K in Gr(k, n)  p_{b}(K) = 0 whenever b is not less than or equal to a} .  (4.5) 
The relevance of the Plücker embedding to Question 4.3 when each
l_{i} equals 1 is seen as follows.
Let L be a (nk)plane, represented as the row space of a
(nk) by n matrix, also written L.
Then a general kplane K meets L nontrivially if and
only if
. 

(4.6) 
Thus the set of kplanes meeting k(nk) general
(nk)planes nontrivially is a complementary linear section
of the Grassmannian, and so the number d(k,n)
of such kplanes is the degree of the Grassmannian in its Plücker
embedding.
More generally, if a in C_{n,k} and
L_{1}, L_{2}, ..., L_{a}
are general (nk)planes, then the number of points in the
intersection of the Schubert varieties