AI 中文总结
本文为单纯射影扇形重心重分的形变锥构造多面体基，将初始光滑边长为1的多面体对应的平坦截断基升级为不可分解多面体基，并探讨其与多种多面体及代数几何对象的关联。
AI 中文摘要
在组合学与代数几何交叉领域研究的若干重要扇形源于其他扇形的重心重分，核心例子是辫群排列（braid arrangement）。标准单形的面构成了辫群排列形变锥的一组基，即广义置换多面体（generalized permutahedra）的锥。我们将这个单纯形基重新解释为标准单形的深度截断集合，并系统地抽象这一视角，为单纯射影扇形重心重分的形变锥构造多面体基。我们进一步证明，这些基可限制为由建筑集（building set）诱导的一系列星型重分得到的扇形形变锥的基。当初始多面体光滑且所有边长均为1时，我们证明可将该基升级为一组平坦截断。随后我们研究辫群排列的两个扩展情形，其中我们能够进一步将平坦截断基升级为不可分解多面体基：1. 标准单形乘积的法扇形的重心重分；2. 辫群排列的重心重分。我们讨论其与阿基米德多面体（Archimedean solids）、正多面体、置换多面体板（permutahedral plates）、根多面体（root polytopes）、置换结合多面体（permutoassociahedra）、简单置换托德多面体（simple permutonestohedra）、宇宙多面体（cosmohedra）、考克斯特置换多面体的全截断（omnitruncations of Coxeter permutahedra）、双置换多面体（bipermutahedra）、π-着色扇形（π-colored fans）以及热带α和β类（tropical α and β classes）的联系与意义。
英文摘要
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 one, 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, $π$-colored fans, and tropical $α$ and $β$ classes.
CommentsMinor revisions. 75 pages and 34 figures
