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# Current Developments in Mathematics

## Volume 2014

### Totally nonnegative Grassmannian and Grassmann polytopes

Pages: 51 – 152

DOI: https://dx.doi.org/10.4310/CDM.2014.v2014.n1.a2

#### Author

#### Abstract

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.

#### Keywords

total nonnegativity, Grassmann polytope, positroid variety, dimers, Grassmann matroid, scattering amplitudes

#### 2010 Mathematics Subject Classification

05C10, 14N15, 52B99

Published 31 December 2015