## Lattice congruences of the weak order.(English)Zbl 1097.20036

Author’s summary: 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 $$\eta_K\colon w\mapsto w_K$$ be the projection onto the parabolic subgroup $$W_K$$. We show that the fibers of $$\eta_K$$ constitute the smallest lattice congruence with $$1\equiv s$$ for every $$s\in(S-K)$$. 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.

### MSC:

 20F55 Reflection and Coxeter groups (group-theoretic aspects) 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 06A07 Combinatorics of partially ordered sets 08A30 Subalgebras, congruence relations

posets; OEIS
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### References:

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