对称群和交错群的容许三重性的几何结构
Geometries admitting trialities for the symmetric and alternating groups
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中文总结 AI 辅助
研究对称群和交错群的容许三重性的几何结构，给出两个无限族，其型保持自同构群同构于\(Sym(n)\)或\(Alt(n)\)，还开发通用方法，且两族剩余结构有差异，极大抛物子群大小增长情况不同。
中文摘要 AI 辅助
在关联几何中，三重性是一种循环交换元素类型三元组的对称性。要求具有三重性的几何结构满足标准正则条件，其构造非常不平凡，已知例子很少。本文给出了两个无限族的旗传递、薄且剩余连通的几何结构，它们容许三重性且无对偶性，其型保持自同构群同构于对称群\(Sym(n)\)或交错群\(Alt(n)\)。还开发了可扩展到其他情形的通用方法。这两个族的剩余结构有根本差异，第一个族中极大抛物子群大小恒定，另一个族中其大小随群的次数基本线性增长。
英文摘要
In incidence geometry, a triality is a symmetry cyclically exchanging triples of types of elements. Requiring geometries with trialities to satisfy standard regularity conditions makes their construction highly non trivial, and known examples are rare. In this paper, we present two infinite families of flag transitive, thin and residually connected geometries admitting trialities and no dualities with type preserving automorphism groups isomorphic to Sym(n) or Alt(n). We also develop general methods that extend to other settings. The residues of the two families are fundamentally different. In the first family, the maximal parabolics have constant size while in the other their size grows essentially linearly with the degree of the group.
