Local coordinates

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Local coordinates

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 Y_aF.(0) and Y_bF.(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 X_a,b be the set of all p by (m+p)-matrices whose entries x_i,j satisfy
           x_{i,a_i} = 1    for i=1, 2, ..., p
           x_i,j  = 0    if j < a_j or j > m + p + 1 - b_p+1-i
If M is a matrix in X_a,b, then the row span of M is a p-plane in the intersection of Y_a F.(0) and Y_b F.(infinity).

   For example, X_{125, 134} consists of all 3 by 7-matrices of the form:

1 x_1,2   x_1,3   x_1,4   0 0 0
0 1 x_2,3 x_2,4 x_2,5   0 0
0 0   0   0   1 x_3,6 x_3,7