Polytopal bases for barycentric subdivisions

Abstract

Several important fans studied at the interface of combinatorics and algebraic geometry arise from barycentric subdivisions of other fans; the central example is the braid arrangement. The collection of faces of the standard simplex forms a basis for the deformation cone of the braid arrangement, i.e. the cone of generalized permutahedra. We reinterpret this simplicial basis as the collection of deep truncations of the standard simplex, and systematically abstract this perspective to produce polytopal bases for the deformation cones of barycentric subdivisions of simplicial projective fans. We further demonstrate that these bases restrict to bases for deformation cones of fans obtained by a sequence of stellar subdivisions induced by a building set. When the starting polytope is smooth with all edge lengths equal to 1, we show that we can upgrade our basis to a collection of flat truncations. We then investigate two extensions of the braid arrangement where we are able to further upgrade our flat truncation basis to an indecomposable polytopal basis: 1.the barycentric subdivision of the normal fan of a product of standard simplices and 2.the barycentric subdivision of the braid arrangement. We discuss connections to, and implications for, Archimedean solids and regular polytopes, permutahedral plates, root polytopes, permutoassociahedra, simple permutonestohedra, cosmohedra, omnitruncations of Coxeter permutahedra, bipermutahedra, $\pi$-colored fans, and tropical $\alpha$ and $\beta$ classes.

Comments are welcome