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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

Software:

posets; OEIS
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References:

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