Abstract
It is an open question to give a combinatorial interpretation of the Falk invariant of a hyperplane arrangement, i.e., the third rank of successive quotients in the lower central series of the fundamental group of the arrangement. In this article, we give a combinatorial formula for this invariant in the case of hyperplane arrangements that are complete lift representations of certain gain graphs. As a corollary, we compute the Falk invariant for the cone of the braid, Shi, Linial, and semiorder arrangements.
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Guo, W., Torielli, M. On the Falk Invariant of Shi and Linial Arrangements. Discrete Comput Geom 66, 751–768 (2021). https://doi.org/10.1007/s00454-020-00266-0
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DOI: https://doi.org/10.1007/s00454-020-00266-0