On the Falk invariant of Shi and Linial arrangements

TL;DR

This paper provides a combinatorial formula for the Falk invariant of certain hyperplane arrangements, including well-known arrangements like braid, Shi, Linial, and semiorder, advancing understanding of their fundamental groups.

Contribution

It introduces a combinatorial formula for the Falk invariant for hyperplane arrangements derived from gain graphs, including notable classes like Shi and Linial arrangements.

Findings

Computed Falk invariant for the cone of the braid, Shi, Linial, and semiorder arrangements.

Established a combinatorial interpretation for the Falk invariant in specific hyperplane arrangements.

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