Abstract
We classify surjective lattice homomorphisms $W\to W'$ between the weak orders on finite Coxeter groups. Equivalently, we classify lattice congruences $\Theta$ on $W$ such that the quotient $W/\Theta$ is isomorphic to $W'$. Surprisingly, surjective homomorphisms exist quite generally: They exist if and only if the diagram of $W'$ is obtained from the diagram of $W$ by deleting vertices, deleting edges, and/or decreasing edge labels. A surjective homomorphism $W\to W'$ is determined by its restrictions to rank-two standard parabolic subgroups of $W$. Despite seeming natural in the setting of Coxeter groups, this determination in rank two is nontrivial. Indeed, from the combinatorial lattice theory point of view, all of these classification results should appear unlikely a priori. As an application of the classification of surjective homomorphisms between weak orders, we also obtain a classification of surjective homomorphisms between Cambrian lattices and a general construction of refinement relations between Cambrian fans.
Highlights
The weak order on a finite Coxeter group W is a partial order structure on W that encodes both the geometric structure of the reflection representation of W and the combinatorial group theory of the defining presentation of W
Recent papers have elucidated the structure of lattice congruences on the weak order [21] and applied this understanding to construct fans coarsening the normal fan of the W -permutohedron [22], combinatorial models of cluster algebras of finite type [23, 25, 30], polytopal realizations of generalized associahedra [9, 10], and sub Hopf algebras of the Malvenuto-Reutenauer Hopf algebra of permutations [14, 15, 17, 22]
The theorem describes the interaction between the combinatorics/geometry of the fan FΘ and the combinatorics of the quotient lattice. (In type A, the fan FΘ is known to be polytopal [19] for any Θ, but no general polytopality result is known in other types.) The fact that a surjective lattice homomorphism η : W → W exists whenever m (r, s) m(r, s) for each pair r, s ∈ S leads to explicit constructions of a fan FΘ coarsening F (W ) such that FΘ is combinatorially isomorphic to the fan F (W ) defined by the reflecting hyperplanes of W
Summary
The weak order on a finite Coxeter group W is a partial order (lattice [2]) structure on W that encodes both the geometric structure of the reflection representation of W and the combinatorial group theory of the defining presentation of W. (In type A, the fan FΘ is known to be polytopal [19] for any Θ, but no general polytopality result is known in other types.) The fact that a surjective lattice homomorphism η : W → W exists whenever m (r, s) m(r, s) for each pair r, s ∈ S leads to explicit constructions of a fan FΘ coarsening F (W ) such that FΘ is combinatorially isomorphic to the fan F (W ) defined by the reflecting hyperplanes of W An example of this geometric point of view (corresponding to Example 1.1) appears as Example 4.5. (See Section 6.1.) Simion’s motivations were not latticetheoretic, so she did not show that the map is a surjective lattice homomorphism She did prove several results that hint at lattice theory, including the fact that fibers of the map are intervals and that the order-theoretic quotient of Bn modulo the fibers of the map is isomorphic to Sn+1. It was Simion’s map that first alerted the author to the fact that interesting homomorphisms exist
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