Normaliser of parabolics in Coxeter groups and hyperplane arrangements from the minimal model program

Abstract

Every Coxeter group gives rise to a hyperplane arrangement, whereby the hyperplanes are the reflection hyperplanes associated to the Coxeter group (viewed as a reflection group). A prototypical example is the symmetric group viewed as a reflection group via the permutation matrices; here, the hyperplanes are exactly those defined by the equation x_i = x_j, one for each transposition (i j). Motivated by the homomological minimal model program (HomMMP), Iyama--Wemyss introduced a new family of hyperplane arrangements called the Tits cone intersections. These are defined by restricting Coxeter arrangements to certain (intersection of) hyperplanes, which produces new arrangements that are no longer defined from Coxeter groups. The goal of this talk is to nonetheless demystify the situation, by relating these arrangements to an old work of Brink--Howlett in 1999 on normaliser of parabolics in Coxeter groups. If time allows, I will speak about the fundamental groups associated to these arrangements, which are really what's important for the HomMMP. This is based on joint work with Owen Garnier, Tony Licata and Oded Yacobi: https://arxiv.org/abs/2509.21915