论文标题
特殊类型的球形建筑中的自动形态和反对，我
Automorphisms and opposition in spherical buildings of exceptional type, I
论文作者
论文摘要
对于球形建筑物的每个自动形态，都有自然关联的“对立图”，该图表编码建筑物简单的类型，这些简单被映射到相反的简单上。如果没有建筑物的腔室（即没有最大的单纯形）被映射到相对的房间上，那么自动形态被称为“家用”。在本文中，我们对类型的分裂球形建筑物的国内自动形态进行完整分类，$ \ Mathsf {e} _6 $，$ \ MATHSF {F} _4 $和$ \ MATHSF {G} _2 $。此外，对于所有分类类型的分裂球形建筑物，我们对（i）国内同源性，（ii）奇瓦利集团标准单位子组的要素以及（iii）自动形态的对立图引起的对立图，最多最多$ 2 $杰出的orbits orbits orbits。我们的结果为长根几何形状中的垂直根长根的长根和产物提供了意想不到的特征，以及在奇瓦利组在任意领域的环境中连接的线性代数基团的密度定理的类似物。
To each automorphism of a spherical building there is naturally associated an "opposition diagram", which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber then the automorphism is called "domestic". In this paper we give the complete classification of domestic automorphisms of split spherical buildings of types $\mathsf{E}_6$, $\mathsf{F}_4$, and $\mathsf{G}_2$. Moreover, for all split spherical buildings of exceptional type we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most $2$ distinguished orbits encircled. Our results provide unexpected characterisations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.
