X. If and Y. ?. , Gr k (C n ) are T -invariant subschemes, and X is X i?j -invariant, then X i?j (X ? Y ) ? X ? X i?j Y

. Hence, is X i?j -invariant, and ? g = ? f ? X [i+1,j]?r , then X i?j ? g ? ? f ? X [i,j?1]?r . Note that this last intersection is a reduced union of interval positroid varieties

{. , 1. ?. {a, and B. , }, based on the choice of (i, j)-mid-sort g. Call the dots in g above S the upper dots of g, and the others the lower dots. First pick letter labels for the upper dots, subject to the requirement that no two dots with the same letter can be NE/SW of one another. In particular it is valid to give every dot a different letter, and it is possible to use only one letter iff g is (i, j)-Vakil. Label all the lower dots with 0. Now project the upper dots to the right

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