Séminaire Lotharingien de Combinatoire, 95B.77 (2026), 12 pp.
Jonathan Boretsky, Veronica Calvo Cortes and Yassine El Maazouz
Totally Positive Skew-Symmetric matrices
Abstract.
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.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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