Basis shape loci and the positive Grassmannian

Abstract

A basis shape locus takes as input data a zero/nonzero pattern in an $n\times k$ matrix, which is equivalent to a presentation of a transversal matroid. The locus is defined as the set of points in the Grassmannian of $k$ planes in ${\mathbb{R}}^n$ which are the row space of a matrix with the prescribed zero/nonzero pattern. We show that this locus depends only on the transversal matroid, not on the specific presentation. When a transversal matroid is a positroid, the closure of its basis shape locus is the associated positroid variety. We give a sufficient, and conjecturally necessary, condition for when a transversal matroid is a positroid. Finally, we discus applications to two programs for computing scattering amplitudes in $\mathcal{N}=4$ SYM theory: one trying to prove that projections of certain positroid cells triangulate the amplituhedron, and another using Wilson loop diagrams.

29 pages, comments welcome

References in corpus (3)

• Total positivity, Grassmannians, and networks
• MHV Diagrams in Momentum Twistor Space
• Schubert calculus and shifting of interval positroid varieties