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The positive-geometry conjecture for Grassmann polytopes and amplituhedra

Let Z:Rn→Rk+mZ:{\mathbb R}^n\to{\mathbb R}^{k+m} be a linear map with n≥k+mn\geq k+m, and let P=Z(Gr(k,n)≥0)P=Z({\mathrm{Gr}}(k,n)_{\geq 0}) be the resulting Grassmann polytope when ZZ is well-defined on the totally nonnegative Grassmannian. When all (k+m)×(k+m)(k+m)\times(k+m) minors of ZZ are positive, this image is the amplituhedron An,k,mA_{n,k,m}. Grassmann-polytope and amplituhedron conjecture. Grassmann polytopes and amplituhedra are positive geometries. The claim is known in the case k=1k=1, where Grassmann polytopes are projective polytopes and the amplituhedron is a cyclic polytope; the general case remains open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The positive-geometry conjecture for Grassmann polytopes and Amplituhedra

    Let a positive geometry be a geometric space equipped with a positive region and a canonical form, and let Grassmann polytopes be images of nonnegative Grassmannians under the maps considered in the paper. Positive-geometry conjecture. Grassmann polytopes, including tree and loop Amplituhedra, are positive geometries. This asserts that these spaces possess the canonical-form structure required of positive geometries, linking the geometry of positive Grassmannians to scattering-amplitude constructions.

    source: Nima Arkani-Hamed, Yuntao Bai and Thomas Lam, “Positive Geometries and Canonical Forms”, arXiv:1703.04541 (2017).

Sources & referencesView supporting material

Primary source

Thomas Lam, “An invitation to positive geometries”, arXiv:2208.05407 (2022).

amplituhedragrassmannianspositive-geometry

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