陪集几何作用于其元素：$j$-对角线扭曲
Coset geometries acting on their elements: The $j$-diagonals twisting
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中文总结 AI 辅助
该研究针对陪集关联系统定义$j$-对角线扭曲构造，证明对应正则超图的存在性，还定义结合Coxeter图的扩张运算，可还原正则多面体的相关扭曲扩张。
中文摘要 AI 辅助
设$\beta$为陪集关联系统并固定类型$j$，我们利用$\beta$的群在不同$j$-元素对上的轨道定义一个Coxeter图，该作用在$j$-元素上诱导出对相应Coxeter群的作用，因此可应用陪集关联系统的扭曲构造，所得陪集几何称为$j$-对角线扭曲；若$\beta$是正则超图，则该$j$-对角线扭曲必为正则超图，据此证明了有限正则超图（其图为树且除一个标号外其余均为4）始终存在，还定义了将该Coxeter图与另一给定Coxeter图结合的扩张运算，对于正则多面体，这些构造可还原McMullen与Schulte的扭曲扩张。
英文摘要
Let $β$ be a coset incidence system and fix a type $j$. We use the orbits of the group of $β$ on pairs of distinct $j$-elements to define a Coxeter graph. The action on the $j$-elements induces an action on the corresponding Coxeter group, so the twisting construction for coset incidence systems can be applied. The resulting coset geometry, called the $j$-diagonals twisting, is always a regular hypertope if $β$ is a regular hypertope. Using this, we show that finite regular hypertope whose diagram is a tree with all but one of the labels equal to four always exist. We also define an extension operation that combines this Coxeter graph with another given Coxeter graph. For regular polytopes, these constructions recover the twisting extensions of McMullen and Schulte.
