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Purity and Separation for Oriented Matroids J. Kevin Thompson Sticking to the same oldLeclerc and Zelevinsky, motivated by the study of quasi commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian. A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of