JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (10): 106-110.doi: 10.6040/j.issn.1671-9352.0.2021.611

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Characteristic polynomials for the generic arrangements of type Dn

ZHANG Miao, QIANG Jing-jing, GAO Rui-mei*   

  1. School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun 130022, Jilin, China
  • Published:2022-10-06

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Abstract: Assume that W is a Weyl group of type Dn in a Euclidean space with dimension n. A generic arrangement of type Dn is one that consists of hyperplanes whose normal vectors are the positive roots of W. It is denoted by A(Dn). To begin with, a one-to-one connection between the signed graphs with no loops and the sub-arrangements of A(Dn) 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(Dn) are given. On this basis, the concrete calculation method of characteristic polynomials of A(Dn) and its sub-arrangements are given.

Key words: generic arrangement of type Dn, signed graph, characteristic polynomial

CLC Number: 

  • O189.1
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