AI 中文总结
本文研究高维单纯形中莫利单纯形为正则的条件，证明正则单纯形为孤立解，并构造反例表明在六维及以上存在无反射对称性的正则莫利单纯形，四维给出非正则精确示例。
AI 中文摘要
将$n$-单纯形的二面角三等分定义了其莫利单纯形。我们研究使该单纯形为正则的原始单纯形。一个基于面法向量格拉姆矩阵的准则将问题归结为矩阵方程。莫利映射在正则单纯形处的导数有两个显式特征值，对于$n\ge3$两者均非零；因此正则单纯形是孤立解。我们猜想在四维和五维中，正则莫利单纯形强制两个超平面反射交换不相交的顶点对。我们证明该结论在每一个维度$6\le n\le200$以及每一个维度$n=\binom k2-1$（其中$k\ge9$）中均不成立：在这些维度中存在具有正则莫利单纯形且无超平面反射对称性的单纯形。对于$8\le n\le200$的示例具有阶为$2(n+1)$的二面体对称性，而无限族基于约翰逊方案。在四维中，我们给出两个非正则示例的精确构造，由次数为$18$和$8$的不可约多项式定义，其伽罗瓦群分别为$S_{18}$和$S_8$。两个示例均不能用根式表达。计算机辅助存在性证明使用精确有理算术和向外舍入的区间。
英文摘要
Trisecting the dihedral angles of an $n$-simplex defines its Morley simplex. We study the original simplices for which this simplex is regular. A criterion in terms of the Gram matrix of the facet normals reduces the problem to a matrix equation. The derivative of the Morley map at the regular simplex has two explicit eigenvalues, both nonzero for $n\ge3$; thus the regular simplex is an isolated solution. We conjecture that in dimensions four and five a regular Morley simplex forces two hyperplane reflections interchanging disjoint pairs of vertices. We prove that this conclusion fails in every dimension $6\le n\le200$ and in every dimension $n=\binom k2-1$ with $k\ge9$: in these dimensions there are simplices with regular Morley simplex and no hyperplane reflection symmetry. The examples for $8\le n\le200$ have dihedral symmetry of order $2(n+1)$, while the infinite family is based on the Johnson scheme. In dimension four we give exact constructions of two nonregular examples, defined by irreducible polynomials of degrees $18$ and $8$ with Galois groups $S_{18}$ and $S_8$. Neither example is expressible by radicals. The computer assisted existence proofs use exact rational arithmetic and intervals with outward rounding.
Comments22 pages
