有限典型空间的对偶极图的次成分I

SUBCONSTITUENTS OF DUAL POLAR GRAPH IN FINITE CLASSICAL SPACE I

设Ｆｑ是一个含有ｑ个元素的有限域.用Г表示Ｆｑ上２ｖ维辛空间的对偶极图．对于Г的任何顶点Ｐ，Г的所有次成分Гｉ（Ｐ）（１≤ｉ≤ｖ）的结构被研究，并且以下结果被证明：Гｉ（Ｐ）同构于［Ｖｉ］ｑ·Ｓｙｍ（ｉ，ｑ），其中Ｓｙｍ（ｉ，ｑ）是Ｆｑ上ｉ×ｉ对称矩阵图。进一步，Ｆｑ上２ｖ＋δ维伪辛空间的对偶极图Ａδ的相应问题也被解决.

Let Fq be a finite field with q elements. Denote by Г the dual polar graph of 2v-dimensional symplectic space over Fq. For any vertex PofГ, the structure of all subconstituents Fi(P) (1 ≤ i≤v) of Г are studied, and it is proved that Гi(P) is isomorphic to [vi]q ·Sym (i, q), where Sym (i, q) is the graph of i × i symmetric matrices over Fq. Moreover, the corresponding problem of the dual polar graph Aδ of (2v+δ)-dimensional pseudo-symplectic space over Fq are also solved.