NDLI: Lattice Congruences of the Weak Order

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Lattice Congruences of the Weak Order

Content Provider	Springer Nature Link
Author	Reading, Nathan
Copyright Year	2005
Abstract	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 η$_{K}$: w↦w$_{K}$ be the projection onto the parabolic subgroup W$_{K}$. We show that the fibers of η$_{K}$ constitute the smallest lattice congruence with 1≡s for every s∈(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.
Starting Page	315
Ending Page	344
Page Count	30
File Format	PDF
ISSN	01678094
Journal	Order
Volume Number	21
Issue Number	4
e-ISSN	15729273
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2005-10-04
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Cambrian lattice congruence uniform Coxeter group parabolic subgroup poset of regions shard simplicial hyperplane arrangement Tamari lattice weak order Theory of Computation Geometry Convex and Discrete Geometry
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Computational Theory and Mathematics Geometry and Topology

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