Positroid Tilings

The positive Grassmannian $${{\,\textrm{Gr}\,}}_{r,n}^{\ge 0}$$ has a beautiful decomposition into positroid cells. Given any surjective map $$\phi : {{\,\textrm{Gr}\,}}_{r,n}^{\ge 0} \rightarrow X$$ , it is natural to try to decompose X using images $$\overline{\phi (S_\pi )}$$ of positroid cells $$S_\pi $$ under $$\phi $$ . If such images have pair-wise disjoint interior, cover X and are full-dimensional, then we call their collection $$\{\overline{\phi (S_\pi )}\}$$ a positroid dissection. If $$\phi $$ is also injective on each cell $$S_\pi $$ , then $$\{\overline{\phi (S_\pi )}\}$$ is a positroid tiling, and each $$\overline{\phi (S_\pi )}$$ is a positroid tile. When $$\phi $$ is the moment map, then $$\overline{\phi (S_\pi )}$$ are positroid polytopes, so a positroid tiling of the hypersimplex is a decomposition into positroid polytopes. When $$\phi $$ is the amplituhedron map, $$\overline{\phi (S_\pi )}$$ are Grasstopes, so they are the building blocks of positroid tilings of the amplituhedron—which provide expressions for $$\mathcal {N}=4$$ SYM scattering amplitudes. In this chapter we prove that T-duality provides a bijection between positroid tilings of the hypersimplex $$\Delta _{k+1,n}$$ and of the $$m=2$$ amplituhedron $$\mathcal {A}_{n,k,2}$$ . We also conjecture the same holds true for positroid dissections. We show how to obtain different types of positroid tilings and dissections from BCFW recursion relations and from the positive tropical Grassmannian $${{\,\textrm{Trop}\,}}^+{{\,\textrm{Gr}\,}}_{k+1,n}$$ —laying the groundwork to realize the tantalizing possibility of a notion of secondary polytope-like structures for Grasstopes.