An n×n skew-symmetric matrix is totally positive exactly when its n(n−1)/2 selected signed minors are positive, and its Pfaffians then follow one universal sign rule.
On two notions of total positivity for generalized partial flag varieties of classical Lie types
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abstract
For Grassmannians, Lusztig's notion of total positivity coincides with positivity of the Plucker coordinates. This coincidence underpins the rich interaction between matroid theory, tropical geometry, and the theory of total positivity. Bloch and Karp furthermore characterized the (type A) partial flag varieties for which the two notions of positivity similarly coincide. We characterize the symplectic (type C) and odd-orthogonal (type B) partial flag varieties for which Lusztig's total positivity coincides with Plucker positivity.
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Totally positive skew-symmetric matrices
An n×n skew-symmetric matrix is totally positive exactly when its n(n−1)/2 selected signed minors are positive, and its Pfaffians then follow one universal sign rule.