Richard P. Stanley Seminar in Combinatorics: Totally positive skew symmetric matrices

A matrix is totally positive if all of its minors are positive. The classical notion of total positivity coincides with the type A version of Lusztig’s more general total positivity in reductive real-split algebraic groups. Since skew-symmetric matrices always have nonpositive entries, they are not totally positive in the classical sense. The space of skew-symmetric matrices is an affine chart of the orthogonal Grassmannian OGr(n, 2n). Thus, we define a skew-symmetric matrix to be totally positive if it lies in the totally positive orthogonal Grassmannian. We provide a positivity criterion for these matrices in terms of a fixed collection of minors, and show that their Pfaffians have a remarkable sign pattern. The totally positive orthogonal Grassmannian is a CW cell complex and is subdivided into Richardson cells. We introduce a method to determine which cell a given point belongs to in terms of its associated matroid.

For information about the Richard P. Stanley Seminar in Combinatorics, visit… https://math.mit.edu/combin/