Snub 24-胞体的四元数表示及其E_8根系的对偶多面体
Quaternionic Representation of Snub 24-Cell and its Dual Polytope Derived From E_8 Root System
摘要：
用具有单位范数的四元数表示四维半正则多面体\textit{snub 24-cell}及其576阶对称群$W(D_{4}):C_{3} $的顶点。它遵循\textbf{$E_{8} $}根系的icosian表示。$H_{4} $的四元根系在$W(F_{4})$和\textbf{$W(H_{4})$}的极大子群之一的\textit{snub 24-cell}的对称群下分裂为24-胞和\textit{snub 24-cell}的顶点.注意，该群同构于$D_{4}$的Weyl群与由$W(D_{4}):C_{3} $表示的3阶循环群的\textbf{}半直积，其Coxeter记法是$[3,4,3^{+}]$。分析了\textit{snub 24-cell}的顶点结构，并将\textbf{$W(H_{4})$}的轨道分解到$W(D_{4}):C_{3} $的轨道下.通过使用群$W(D_{4}):C_{3} $的子群，用四元数明确地分析了snub 24-cell的细胞结构。特别地，证明了在群$W(D_{4}):C_{3} $下，具有120个顶点的对偶多面体600-cell和具有600个顶点的对偶多面体120-cell分别分解为120=24+96和600=24+96+192+288.明确构造了\textit{snub 24-cell}的对偶多面体。附录中给出了阿基米德$W(H_{4})$多面体在群$W(D_{4}):C_{3} $对称性下的分解。
英文摘要：
Vertices of the 4-dimensional semi-regular polytope, \textit{snub 24-cell} and its symmetry group $W(D_{4}):C_{3} $ of order 576 are represented in terms of quaternions with unit norm. It follows from the icosian representation of \textbf{$E_{8} $} root system. The quaternionic root system of $H_{4} $ splits as the vertices of 24-cell and the \textit{snub 24-cell} under the symmetry group of the \textit{snub 24-cell} which is one of the maximal subgroups of the group \textbf{$W(H_{4})$} as well as $W(F_{4})$. It is noted that the group is isomorphic to the\textbf{}semi-direct product of the Weyl group of $D_{4}$ with the cyclic group of order 3 denoted by $W(D_{4}):C_{3} $, the Coxeter notation for which is $[3,4,3^{+}]$. We analyze the vertex structure of the \textit{snub 24-cell} and decompose the orbits of \textbf{$W(H_{4})$} under the orbits of $W(D_{4}):C_{3} $. The cell structure of the snub 24-cell has been explicitly analyzed with quaternions by using the subgroups of the group $W(D_{4}):C_{3} $. In particular, it has been shown that the dual polytopes 600-cell with 120 vertices and 120-cell with 600 vertices decompose as 120=24+96 and 600=24+96+192+288 respectively under the group $W(D_{4}):C_{3} $. The dual polytope of the \textit{snub 24-cell} is explicitly constructed. Decompositions of the Archimedean $W(H_{4})$ polytopes under the symmetry of the group $W(D_{4}):C_{3} $ are given in the appendix.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
