D_(n)型通有构形的特征多项式

Characteristic polynomials for the generic arrangements of type Dn

设W是n维欧氏空间中的D型Weyl群,以W的正根为法向量的超平面形成的通有构形称为D型通有构形,记为A(D)。首先建立了不含自环的符号图与A(D)的子构形的一一对应关系;其次,研究了一个符号圈线性相关的充要条件;最后从符号图的角度给出A(D)的子构形线性无关的充要条件。在此基础上,给出A(D)及其子构形的特征多项式的具体计算方法。

Assume that W is a Weyl group of type Din a Euclidean space with dimension n. A generic arrangement of type Dis one that consists of hyperplanes whose normal vectors are the positive roots of W. It is denoted by A(D). To begin with, a one-to-one connection between the signed graphs with no loops and the sub-arrangements of A(D) is established. Then, the sufficient and necessary condition for the linear dependence of signed circles are studied. In the end, from the point of view of signed graphs, the necessary and sufficient conditions for linear independence of sub-arrangements of A(D) are given. On this basis, the concrete calculation method of characteristic polynomials of A(D) and its sub-arrangements are given.