Totally positive skew-symmetric matrices

by Veronica Calvo Cortes (MPI for Mathematics in the Sciences, Leipzig)

Thursday 21 Nov 2024, 14:00 → 15:00 Europe/Berlin

A.1.01/03 - Alps (New)

Classically, totally positive matrices are matrices with all positive minors. This notion coincides with Lusztig's more general notion of positivity in the case of GL(n). A skew-symmetric matrix can never satisfy this condition. However, the space of skew-symmetric matrices is an affine chart of the orthogonal Grassmannian OGr(n,2n). Thus, we define a skew-symmetric matrix as 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. Positivity notions in real reductive groups besides GL(n) have become increasingly important to understand as positroid combinatorics are gaining more relevance in the study of scattering amplitudes in quantum field theory.