## Abstract

The positive Grassmannian [FIGURE] is a cell complex consisting of all points in the real Grassmannian whose Plücker coordinates are non-negative. In this paper we consider the image of the positive Grassmannian and its positroid cells under two different maps: the moment map μ onto the hypersimplex [31] and the amplituhedron map ˜Z onto the amplituhedron [6]. For either map, we define a positroid dissection to be a collection of images of positroid cells that are disjoint and cover a dense subset of the image. Positroid dissections of the hypersimplex are of interest because they include many matroid subdivisions; meanwhile, positroid dissections of the amplituhedron can be used to calculate the amplituhedron’s ‘volume’, which in turn computes scattering amplitudes in N = 4 super Yang-Mills. We define a map we call T-duality from cells of [FIGURE] to cells of [FIGURE] and conjecture that it induces a bijection from positroid dissections of the hypersimplex
_{k}
_{+1,}
_{n} to positroid dissections of the amplituhedron A
_{n}
_{k}
_{,2}; we prove this conjecture for the (infinite) class of BCFW dissections. We note that T-duality is particularly striking because the hypersimplex is an (n − 1)-dimensional polytope while the amplituhedron A
_{n}
_{k}
_{,2} is a 2k-dimensional non-polytopal subset of the Grassmannian Gr
_{k}
_{k}
_{+2}. Moreover, we prove that the positive tropical Grassmannian is the secondary fan for the regular positroid subdivisions of the hypersimplex, and prove that a matroid polytope is a positroid polytope if and only if all 2D faces are positroid polytopes. Finally, toward the goal of generalizing T-duality for higher m, we define the momentum amplituhedron for any even m.

Original language | English |
---|---|

Article number | rnad010 |

Pages (from-to) | 16778-16836 |

Number of pages | 59 |

Journal | International Mathematical Research Notices |

Volume | 2023 |

Issue number | 19 |

Early online date | 7 Mar 2023 |

DOIs | |

Publication status | Published - 1 Oct 2023 |