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...Schubertcalculusandpuzzles notes for the osaka summer school allenknutson contents interval positroid varieties schubert calculus first positivity result rank vakil s littlewood richardson rule combinatorial shifting geometricshifting degeneration order partial puzzles equivariant and k extensions homology cohomology theory otherpartial ag manifolds references in my rst lecture i ll present a family of interpolating between called denitions mk n nmatricesoverc stiefel manifold m is open subset which rows are linearly dependent grassmanniangrk c gl nmk isthespaceofk planesinc twomatrices give same plane if they re related by row operations to kill that ambiguity put things reduced echelonform fromtherewecanassociate date draft july discrete invariant bit string like with pivot positions remaining careful not reverse let denote set such strings examples most matrices mrank all columns zero except last kth column span but there no other dependen cies proposition decomposition grassmannian...