Alphabeta MathResearch collection / 01
Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections
Owen Garnier, Edmund Heng, Anthony Licata, Oded Yacobi
Let (W, S) be a Coxeter system and let J ⊆S. Motivated by 3-fold flops, Iyama and Wemyss study the hyperplane arrangement in the Tits cone intersection of J, which is a J- relative generalisation of the classical Coxeter arrangement. For Γ of spherical type, the authors show that the complexified complement of this J-relative hyperplane arrangement is a K(π, 1) space for the normaliser (quotient) of the standard parabolic subgroup attached to J in the Artin–Tits group.
For general Γ the authors show that the Brink–Howlett groupoid, which describes normalisers of parabolic subgroups of the Coxeter groups, has its universal cover described by the wall-and-chamber structure of the Tits cone intersection. The authors use this to show that wall crossing sequences satisfy an “atomic Matsumoto relation”, generalising a theorem of Ko and answering questions raised by Iyama and Wemyss.
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