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 representation of certain gain graphs. As a corollary, we compute the Falk invariant for the cone of the braid, Shi, Linial and semiorder arrangements.

中文翻译：

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关于 Shi 和 Linial 排列的 Falk 不变量

给出超平面排列的 Falk 不变量的组合解释是一个悬而未决的问题，即排列的基本群的下中心级数中连续商的第三阶。在本文中，我们在超平面排列是某些增益图的完整提升表示的情况下给出了该不变量的组合公式。作为推论，我们计算了辫子、Shi、Linial 和半阶排列的锥体的 Falk 不变量。