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3.iii. Local coordinates for the intersection of 2 Schubert varieties

Given two sequences a and b parameterizing Schubert varieties, we define local coordinates for the set of p-planes H which lie in the intersection of Schubert varieties YaF.(0) and YbF.(infinity), where F.(s) is the flag of subspaces osculating the standard rational normal curve at s. Its k-plane is the row space of the following k by (m+p)-matrix:

   Let Xa,b be the set of all p by (m+p)-matrices whose entries xi,j satisfy
           xi,ai = 1    for i=1, 2, ..., p
           xi,j  = 0    if  j < aj  or   j > m + p + 1 - bp+1-i
If M is a matrix in Xa,b, then the row span of M is a p-plane in the intersection of Ya F.(0) and Yb F.(infinity).

   For example, X125, 134 consists of all 3 by 7-matrices of the form:
 1  x1,2   x1,3   x1,4   0  0   0
 0  1 x2,3 x2,4 x2,5    0  0
 0   0   0   0   1 x3,6 x3,7


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