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Assume you are given two subspace $V$ and $W$, belonging to some Schubert cells $C_I$ and $C_J$. Is there an elementary closed form description of $V\cap W$ in $C_{I\cap J}$?

Here by "elementary" I mean everything is with respect to the standard flag, and thus the subspaces can be thought as column space of matrices of appropriate size, and $I, J$ are subsets of row/column indices.

Again, I am not looking for a description that involves, say, algebraic geometry language or other alikes.

user 1987
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    What do you mean by $C_{I \cap J}$ ? And shouldn't $V \cap W$ depend on $V$ and $W$, not only on $C_I$ and $C_J$ ? – darij grinberg Jul 27 '19 at 17:21
  • Yes, of course it will! And by $C_{I\cap J}$ I mean the cell that contains the intersection, though I now realize the confusion. I am simply asking for a description of the cell that contains $V\cap W$. – user 1987 Aug 04 '19 at 09:34

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