Equations for Schubert varieties

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3.iv. Equations for Schubert varieties

Let X be a p by (m+p)-matrix, which we think of as coordinates for p-planes and F. a complete flag. Then the condition that X lie in Y_aF., (the intersection of X with F_m+p+1-a_i has dimension at least i), is
(m+p+1-i-a_i)-minors of

for each i=1,2,...,p.
Observe that in general, this gives many more equations than the codimension |a| of the Schubert variety Y_a F.. Using these equations and a choice of local coordinates, we may convert the geometric statement of Conjecture 2 into a statement about certain polynomial systems, which will in general be overdetermined.