delta-拟阵多面体的正则二进三角剖分
Regular dyadic triangulations of delta-matroid polytopes
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中文总结 AI 辅助
本文证明每个delta-拟阵多面体（及更一般的整B型广义置换多面体）存在正则二进三角剖分，其最大单纯形体积为2的幂，并指出B型中唯一非幺模单纯形例外，基于B型根配置的完全二进性。
中文摘要 AI 辅助
Backman和Liu证明了每个A型整广义置换多面体，特别是每个拟阵基多面体，都允许一个正则幺模三角剖分。类似命题在B型中不成立：delta-拟阵单纯形 conv{0, e1+e2, e1+e3, e2+e3} 的归一化体积为2，且除顶点外没有格点，因此它没有幺模三角剖分。我们进一步证明，在0/1立方体的自然对称和删除常数坐标的意义下，它是唯一的非幺模且为单纯形的delta-拟阵多面体。我们转而证明每个delta-拟阵多面体都允许一个正则二进三角剖分，即其最大单纯形的归一化体积为2的幂的格三角剖分。更一般地，每个整B型广义置换多面体都允许这样的三角剖分。主要的格论成分是B型根配置构成一个完全二进系统，这是完全幺模性的2-局部类比。作为推论，这些多面体满足整数分解性质的二进版本。在每个维度中，相应的指数可以统一选取，即使对于delta-拟阵多面体，普通整数分解可能失败。
英文摘要
Backman and Liu proved that every integral generalized permutohedron of type $A$, and in particular every matroid base polytope, admits a regular unimodular triangulation. The analogous statement fails in type $B$: the delta-matroid simplex \[\operatorname*{conv}\{\mathbf{0},\ e_1+e_2,\ e_1+e_3,\ e_2+e_3\}\] has normalized volume $2$ and no lattice points other than its vertices, so it has no unimodular triangulation. We show moreover that, up to the natural symmetries of the $0/1$ cube and deletion of constant coordinates, it is the unique non-unimodular delta-matroid polytope that is a simplex. We prove instead that every delta-matroid polytope admits a regular dyadic triangulation, meaning a lattice triangulation whose maximal simplices have normalized volumes that are powers of two. More generally, every integral type $B$ generalized permutohedron admits such a triangulation. The main lattice-theoretic ingredient is that the type $B$ root configuration forms a totally dyadic system, a $2$-local analogue of total unimodularity. As a consequence, these polytopes satisfy a dyadic version of the integer decomposition property. In each dimension the corresponding exponent can be chosen uniformly, even though ordinary integer decomposition can fail for delta-matroid polytopes.
发表机构
- Université libre de Bruxelles（布鲁塞尔自由大学）
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