, Volume 21, Issue 4, pp 315–344 | Cite as

Lattice Congruences of the Weak Order

  • Nathan ReadingEmail author


We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of join-irreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset \(K\subseteq S\) , let ηK: wwK be the projection onto the parabolic subgroup WK. We show that the fibers of ηK constitute the smallest lattice congruence with 1≡s for every s∈(SK). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of join-irreducibles of the congruence lattice of the weak order.


Cambrian lattice congruence uniform Coxeter group parabolic subgroup poset of regions shard simplicial hyperplane arrangement Tamari lattice weak order 


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© Springer 2005

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of MichiganAnn ArborUSA

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