Current Developments in Mathematics
Totally nonnegative Grassmannian and Grassmann polytopes
Pages: 51 – 152
These are lecture notes intended to supplement my second lecture at the Current Developments in Mathematics conference in 2014. In the first half of article, we give an introduction to the totally nonnegative Grassmannian together with a survey of some more recent work. In the second half of the article, we give a definition of a Grassmann polytope motivated by work of physicists on the amplituhedron. We propose to use Schubert calculus and canonical bases to replace linear algebra and convexity in the theory of polytopes.
total nonnegativity, Grassmann polytope, positroid variety, dimers, Grassmann matroid, scattering amplitudes
2010 Mathematics Subject Classification
05C10, 14N15, 52B99