On the Falk Invariant of Shi and Linial Arrangements

Weili Guo; Michele Torielli

doi:10.1007/s00454-020-00266-0

On the Falk Invariant of Shi and Linial Arrangements

Research output: Contribution to journal › Article › peer-review

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.

Original language	English (US)
Pages (from-to)	751-768
Number of pages	18
Journal	Discrete and Computational Geometry
Volume	66
Issue number	2
DOIs	https://doi.org/10.1007/s00454-020-00266-0
State	Published - Sep 2021
Externally published	Yes

Keywords

Falk invariant
Gain graph
Hyperplane arrangement
Lower central series
Shi arrangement

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1007/s00454-020-00266-0

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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.",

keywords = "Falk invariant, Gain graph, Hyperplane arrangement, Lower central series, Shi arrangement",

author = "Weili Guo and Michele Torielli",

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AB - 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.

KW - Falk invariant

KW - Gain graph

KW - Hyperplane arrangement

KW - Lower central series

KW - Shi arrangement

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On the Falk Invariant of Shi and Linial Arrangements

Abstract

Keywords

ASJC Scopus subject areas

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