Dual polar spaces
Dual polar spaces
复制标题
双极空间
DOI：
10.1007/bf00147332
复制
发表时间：
1982
影响因子：
0.5
通讯作者：
P. Cameron
中科院分区：
文献类型：
--
作者：
P. Cameron
Buekenhout和Shult [2]给出了一组简单的公理，用点和线来刻画极空间。鉴于建筑物和它的Coxeter复形之间的关系（见Tits [10]），这些点-线几何可能被称为“广义八面体”（比较射影空间作为“广义单形”）。这是本文的目的公理化对偶极空间（“广义立方体”）。换句话说，我们在定理1中给出了点和线的几何上的一个充要条件，使点与极空间的极大子空间相同，线是包含次极大子空间的极大子空间的集合。这似乎有两个原因。首先，与秩为n的对偶极空间相关联的几何的直径为n，而任何极空间的秩为2。第二，对偶极空间的“积木”是广义四边形，子空间是秩较小的对偶极空间;在极空间中，所有子空间都是射影的。在有限情形下，定理3给出了主定理的另一个版本，用数值条件代替了关联公理。这包括Dembowski和瓦格纳[5]、Shult和Yanushka [9]以及卡梅隆和Drake [3]定理的特殊情况。最后，在定理4中，我们用自同构群的传递性代替其中的一个条件，得到了一类距离传递图的一个由它们的交阵刻画的特征。这包括Kantor [6]定理的一个特例。
Buekenhout and Shult [2] gave a simple set of axioms characterising polar spaces in terms of their points and lines. In view of the relation between a building and its Coxeter complex (see Tits [10]), these point-line geometries might be called'generalised octahedra'(compare projective spaces as' generalised simplexes'). It is the purpose of this paper to axiomatise dual polar spaces ('generalised cubes'). In other words, we give in Theorem 1 a necessary and sufficient condition on a geometry of points and lines for the points to be identified with the maximal subspaces of a polar space, a line being the set of maximal subspaces containing a next-to-maximal subspace.It will be noted that our axioms are somewhat more complicated than those of Buekenhout and Shult. There seem to be two reasons for this. First, the geometry associated with a dual polar space of rank n has diameter n, whereas any polar space has rank 2. Second, the'building blocks' of dual polar spaces are generalised quadrangles, and the subspaces are dual polar spaces of smaller rank; in a polar space, all subspaces are projective. In the finite case, another version of the main theorem is given in Theorem 3, with numerical conditions in place of the incidence axioms. This includes special cases of theorems of Dembowski and Wagner [5], Shult and Yanushka [9], and Cameron and Drake [3]. Finally, in Theorem 4, we replace one of these conditions by a transitivity property of the automorphism group, obtaining a characterisation of a class of distance-transitive graphs by their intersection arrays. This includes a special case of a theorem of Kantor [6].