Lattice congruences of the weak order

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 of S, let η_K:w \mapsto 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\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.

26 pages, 4 figures

References in corpus (4)

• Cambrian Lattices
• Lattice congruences, fans and Hopf algebras
• The Order Dimension of the Poset of Regions in a Hyperplane Arrangement
• Tamari lattices and noncrossing partitions in type B and beyond

Cited by in corpus (41)

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• Cambrian Lattices
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• Lattice congruences, fans and Hopf algebras
• Brick polytopes of spherical subword complexes and generalized associahedra
• Sortable elements in infinite Coxeter groups
• The Hopf algebra of diagonal rectangulations
• Associahedra via spines
• Non-kissing complexes and tau-tilting for gentle algebras
• Noncrossing partitions and the shard intersection order
• Brick polytopes, lattice quotients, and Hopf algebras
• Lattice structure of Weyl groups via representation theory of preprojective algebras
• The facial weak order and its lattice quotients
• Compatibility fans for graphical nested complexes
• Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
• The weak order on integer posets
• Celebrating Loday's Associahedron
• Removahedral congruences versus permutree congruences
• The associative operad and the weak order on the symmetric groups
• A combinatorial approach to scattering diagrams
• Signed tree associahedra
• Shard polytopes
• A broad class of shellable lattices
• Acyclic reorientation lattices and their lattice quotients
• The fundamental theorem of finite semidistributive lattices
• Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra
• Noncrossing arc diagrams and canonical join representations
• The weak order on Weyl posets
• Cambrian triangulations and their tropical realizations
• Coxeter-biCatalan combinatorics
• Lattice homomorphisms between weak orders
• The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure
• Geometric realizations of the $s$-weak order and its lattice quotients
• Hopf structures on the multiplihedra
• Derived Semidistributive Lattices
• Structural Properties of the Cambrian Semilattices -- Consequences of Semidistributivity
• Sublattices of associahedra and permutohedra
• Permutree sorting
• Interval hypergraphic lattices
• Tamari lattices, forests and Thompson monoids