AI 中文总结
本文针对有限域$\text{FF}_q$上的模式群构造自然陪集几何，证明其几何性质，建立闭集组合特征与抛物子群性质的对应，刻画自同构并研究域变化的影响，丰富了群论与几何关联的研究。
AI 中文摘要
陪集关联几何是连接群论与几何的重要工具。尽管单位上三角群及其模式子群的表示已被广泛研究，但这些群的底层几何结构却大多未被探索。本文中，我们针对有限域$\boldsymbol{\text{FF}}_q$上的每个模式群构造对应的自然陪集几何，并证明这些结构的几何性质。我们展示了其定义闭集的组合特征如何直接对应于抛物子群的性质，特别是抛物子群的交集以及Borel子群的正规性。最后，我们刻画这些几何的自同构，并研究在同一特征$p$下域的变化时模式几何的表现。
英文摘要
Coset incidence geometries are an important tool which connect group theory and geometry. While the representations of the unitriangular group and its pattern subgroups have been studied extensively, the underlying geometric structures of these groups has remained largely unexplored. In this article, we construct the natural coset geometry associated with each pattern group over a finite field $\FF_q$, and prove geometrical properties of theses structures. We show how combinatorial features of their defining closed sets directly correspond to properties of their parabolic subgroups, in particular the intersection of parabolics and normality of the Borel subgroup. Finally, we characterize the automorphisms of these geometries and examine how pattern geometries behave under change of field within the same characteristic $p$.
