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The Global Invariant of Signed Graphic hyperplane Arrangements

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Abstract

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an interesting and complicated invariant. The third rank of successive quotients in the lower central series of the fundamental group was called the global invariant of the arrangement by Falk. Falk gave a general formula to compute the global invariant, and asked for a combinatorial interpretation of the global invariant. Schenck and Suciu proved that the global invariant of a graphic arrangement is double of the number of cliques with three or four vertices in the graph with which the arrangement associated. This solved Falk’s problem in the case of graphic arrangements. While in the case of signed graphic arrangements, we obtained a similar combinatorial formula. In this paper, we give a direct and simple proof for this combinatorial formula.

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Authors and Affiliations

  1. School of Science, Beijing University of Chemical Technology, Beijing, 100029, People’s Republic of China

    Qiumin Guo, Weili Guo, Wentao Hu & Guangfeng Jiang

Authors
  1. Qiumin Guo
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  2. Weili Guo
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  3. Wentao Hu
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  4. Guangfeng Jiang
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Correspondence to Qiumin Guo.

Additional information

Supported by NSF of China under Grant 11071010.

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Guo, Q., Guo, W., Hu, W. et al. The Global Invariant of Signed Graphic hyperplane Arrangements. Graphs and Combinatorics 33, 527–535 (2017). https://doi.org/10.1007/s00373-017-1770-y

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  • DOI: https://doi.org/10.1007/s00373-017-1770-y

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