JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2018, Vol. 53 ›› Issue (6): 70-75.doi: 10.6040/j.issn.1671-9352.0.2017.644

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Freeness of arrangements between the Weyl arrangements of types An-1 and Bn

GAO Rui-mei, CHU Ying*   

  1. Department of Science, Changchun University of Science and Technology, Changchun 130022, Jilin, China
  • Received:2017-12-19 Online:2018-06-20 Published:2018-06-13

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Abstract: The set of reflecting hyperplanes of a Weyl group is called Weyl arrangement. Assume the notations An-1 and Bn denote the Weyl arrangements of types An-1 and Bn respectively. The arrangement A which satisfies An-1⊂A⊂Bn is called the arrangement between An-1 and Bn. Firstly, we study threshold graphs, and give a construction for threshold graphs. Secondly, we study the freeness of the arrangements between An-1 and Bn by using threshold graphs. And we conclude: for any integer k satisfying |An-1|n|, there exists a free arrangement between An-1 and Bn with cardinality k. A similar conclusion for the non-freeness arrangements between An-1 and Bn is given.

Key words: Weyl arrangement, threshold graph, signed graph, graphical arrangement, freeness

CLC Number: 

  • O189
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