Totally positive skew-symmetric matrices

TL;DR

This paper extends the concept of total positivity to skew-symmetric matrices by defining and characterizing totally positive skew-symmetric matrices within the orthogonal Grassmannian, revealing their structure and cell decomposition.

Contribution

It introduces a new notion of total positivity for skew-symmetric matrices via the orthogonal Grassmannian and provides criteria and cell decomposition methods.

Findings

A positivity criterion based on minors for skew-symmetric matrices.

Pfaffians of these matrices exhibit a specific sign pattern.

The orthogonal Grassmannian forms a CW cell complex subdivided into Richardson cells.

Abstract

A matrix is totally positive if all of its minors are positive. This notion of 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 $\mathrm{OGr}(n,2n)$. 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.