Lattice homomorphisms between weak orders

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.


Introduction
The weak order on a finite Coxeter group W is a partial order (in fact, 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 . 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]. A thorough discussion of lattice congruences of the weak order (and more generally of certain posets of regions) is available in [27,28].
The purpose of this paper is to classify surjective lattice homomorphisms between the weak orders on two finite Coxeter groups W and W ′ . Equivalently, we classify the lattice congruences Θ on a finite Coxeter group W such that the quotient lattice W/Θ is isomorphic to the weak order on a finite Coxeter group W ′ .
From the point of view of combinatorial lattice theory, the classification results are quite surprising a priori. As an illustration of the almost miraculous nature of the situation, we begin this introduction with a representative example (Example 1.1), after giving just enough lattice-theoretic details to make the example understandable. (More lattice-theoretic details are in Section 4.) A homomorphism from a lattice L to a lattice L ′ is a map η : L → L ′ such that η(x ∧ y) = η(x) ∧ η(y) and η(x ∨ y) = η(x) ∨ η(y). A congruence on a lattice L is an equivalence relation such that (x 1 ≡ y 1 and x 2 ≡ y 2 ) implies [(x 1 ∧ x 2 ) ≡ (y 1 ∧ y 2 ) and (x 1 ∨ x 2 ) ≡ (y 1 ∨ y 2 )] .
Given a congruence Θ on L, the quotient lattice L/Θ is a lattice structure on the set of equivalence classes where the meet C 1 ∧ C 2 of two classes is the equivalence class containing x ∧ y for any x ∈ C 1 and y ∈ C 2 and the join is described similarly. When L is a finite lattice, the congruence classes of any congruence Θ on L are intervals. The quotient L/Θ is isomorphic to the subposet of L induced by the set of elements x such that x is the bottom element of its congruence class.
We use the symbol ⋖ for cover relations in L and often call a pair x ⋖ y an edge (because it forms an edge in the Hasse diagram of L). If x ⋖ y and x ≡ y, then we say that the congruence contracts the edge x ⋖ y. Since congruence classes on a finite lattice are intervals, to specify a congruence it is enough to specify which edges the congruence contracts. Edges cannot be contracted independently; rather, contracting some edge typically forces the contraction of other edges to ensure that the result is a congruence. Forcing among edge contractions on the weak order is governed entirely 1 by a local forcing rule in polygons. A polygon is an interval such that the underlying graph of the Hasse diagram of the interval is a cycle. There are two top edges in a polygon, the two that are incident to the maximum, and two bottom edges, incident to the minimum. The remaining edges in the interval, if there are any, are side edges. The forcing rule for polygons is the following: if a top (respectively bottom) edge is contracted, then the opposite bottom (respectively top) edge must also be contracted, and all side edges (if there are any) must be contracted. One case of the rule is illustrated in Figure 1 The Coxeter diagram of a Coxeter group of type A 3 has the same diagram, except that the label 4 is replaced by 3. Informally, we can turn the picture in Figure 2.a into the analogous picture for A 3 , by contracting two side edges of the octagon to form a hexagon, as indicated by shading in Figure 2.b. If we take the same two edges in the whole weak order on W , we can use the polygonal forcing rules to find the finest lattice congruence that contracts the two edges. This congruence is illustrated in Figure 3.a. A priori, we shouldn't expect this congruence to have any significance, but, surprisingly, the quotient, shown in Figure 3.b, of the weak order modulo this congruence is isomorphic to the weak order on a Coxeter group of type A 3 . (Recall from above that the lattice quotient is isomorphic to the subposet consisting of elements that are at the bottom of their congruence class.) To recap: We start with the weak order on B 3 , look at some polygons at the bottom of the weak order that encode the Coxeter diagram for B 3 , and naïvely contract edges of these polygons to make one of the polygons smaller so that the polygons instead encode the Coxeter diagram for A 3 . Miraculously, the contracted edges generate a congruence such that the quotient is the weak order on A 3 .
In general, a diagram homomorphism starts with the Coxeter diagram of a Coxeter system (W, S), deletes vertices, decreases labels on edges, and/or erases edges, and relabels the vertices to obtain the Coxeter diagram of some Coxeter system (W ′ , S ′ ). (When no vertices are deleted, no labels are decreased, and no edges are erased, this is a diagram isomorphism and when (W ′ , S ′ ) = (W, S) it is a diagram automorphism.) For brevity in what follows, we will say "a diagram homomorphism from (W, S) to (W ′ , S ′ )" to mean "a diagram homomorphism from the Coxeter diagram of (W, S) to the Coxeter diagram of (W ′ , S ′ )." The first main results of the paper are the following theorem and several more detailed versions of it. Remark 1.4. The existence of surjective homomorphisms between weak orders is not difficult to prove, a posteriori: We give explicit maps. However, the other results in the classification need more machinery, specifically the machinery of shards, as explained in Section 4. Furthermore, without the machinery of shards, we would not be able to find the explicit homomorphisms that prove existence.
In order to make more detailed classification statements, we first give a factorization result for any surjective lattice homomorphism between weak orders. Given a finite Coxeter group W and a standard parabolic subgroup W J , the parabolic homomorphism η J is the map taking w ∈ W to w J ∈ W J , where w J is the parabolic factor in the usual factorization of w as an element of the parabolic subgroup times an element of the quotient. A parabolic homomorphism corresponds to a diagram homomorphism that only deletes vertices from the diagram of W .
An atom in a finite lattice is an element that covers the minimal element0. We will call a homomorphism of finite lattices compressive if it is surjective and restricts to a bijection between the sets of atoms of the two lattices. (The term is an analogy to the physical process of compression where atoms are not created or destroyed but are brought closer together. If η : L → L ′ is compressive, then η moves two atoms a 1 , a 2 of L weakly closer in the sense that the interval below η(a 1 )∨η(a 2 ) has weakly fewer elements than the interval below a 1 ∨a 2 .) In particular, a compressive homomorphism between weak orders on Coxeter systems (W, S) and (W ′ , S ′ ) is a surjective homomorphism W → W ′ that restricts to a bijection between S and S ′ . In Section 2, we prove the following theorem. Parabolic homomorphisms and their associated congruences are well understood. (See [21,Section 6].) The task, therefore, becomes to understand compressive homomorphisms between Coxeter groups. To study compressive homomorphisms from (W, S) to (W ′ , S ′ ), we may as well take S ′ = S and require η to restrict to the identity on S. For each r, s ∈ S, let m(r, s) be the order of rs in W , and let m ′ (r, s) be the order of rs in W ′ . Elementary considerations show that if η : W → W ′ is compressive, then m ′ (r, s) ≤ m(r, s) for each pair r, s ∈ S. (See Proposition 2.6.) Thus a compressive homomorphism corresponds to a diagram homomorphism that only erases edges from and/or reduces edge labels on the diagram of W . More surprising, this property is sufficient to guarantee the existence of a compressive homomorphism. The following theorem shows that Example 1.1 is typical, rather than unusual. Together with Theorem 1.5, it implies Theorem 1.2 and adds additional detail. Theorem 1.6. Suppose (W, S) and (W ′ , S) are finite Coxeter systems. Then there exists a compressive homomorphism from W to W ′ , fixing S, if and only if m ′ (r, s) ≤ m(r, s) for each pair r, s ∈ S. If so, then the homomorphism can be chosen so that the associated congruence on W is homogeneous of degree 2.
We will review the definition of homogeneous congruences in Section 4. Informally, a homogeneous congruence of degree 2 is a congruence that is determined by contracting edges located in the order ideal of the weak order that describes the diagram of W , as in Figure 2. The situation is perhaps best appreciated by analogy: Showing that W ′ is isomorphic to the quotient of W modulo a homogeneous congruence of degree 2 is analogous to finding that a graded ring R is isomorphic to another graded ring R ′ modulo an ideal generated by homogeneous elements of degree 2. It should be noted, however, that lattice congruences are in general more complicated than ring congruences because the classes of a lattice congruence are not in general defined by an ideal. See [1,.
Given (W, S) and (W ′ , S) with m ′ (r, s) ≤ m(r, s) for each pair r, s ∈ S, there may be several homomorphisms from W to W ′ whose associated congruence is homogeneous of degree 2. There may also be several homomorphisms whose congruence is not homogeneous. (In these non-homogeneous cases, the degree of the congruence is always 3.) These several possibilities are well-characterized, as we now explain. Elementary considerations show that the restriction of a compressive homomorphism to any standard parabolic subgroup is still compressive. (See Proposition 2.6.) It turns out that compressive homomorphisms of Coxeter groups are determined by their restrictions to rank-two standard parabolic subgroups. As will be apparent in Section 5, for each pair {r, s} ⊆ S with r = s, there are exactly a b 2 choices of η {r,s} , where a = m(r, s) − 2 and b = m(r, s) − m ′ (r, s).
In Example 1.1, there are four ways to choose all of the maps η {r,s} : For both pairs r, s with m(r, s) ≤ 3, we must choose the identity map. For the pair r, s with m(r, s) = 4 and m ′ (r, s) = 3, there are four choices, corresponding to the four ways to contract one "left" side edge and one "right" side edge in the octagonal interval of Figure 2.a. Theorem 1.7 says, in particular, that in the example there are at most four homomorphisms that fix S pointwise.
Combining Theorem 1.7 with Theorem 1.5 leads immediately to the following more general statement. Corollary 1.8. Let W and W ′ be finite Coxeter groups. A surjective homomorphism from the weak order on W to the weak order on W ′ is determined by its restrictions to rank-two standard parabolic subgroups.
The statement of Theorem 1.7 on the uniqueness of compressive homomorphisms, given their restrictions to rank-two standard parabolic subgroups, is remarkably close to being an existence and uniqueness theorem, in the sense that the phrase "at most one" can almost be replaced with "exactly one." The only exceptions arise when W has a standard parabolic subgroup of type H 3 such that the corresponding standard parabolic subgroup of W ′ is of type B 3 . In particular, adding the hypothesis that W and W ′ are crystallographic turns Theorem 1.7 into an existence and uniqueness theorem (stated as Theorem 7.2). Less generally, when W and W ′ are simply laced meaning that all edges in their diagrams are unlabeled), the existence and uniqueness theorem holds, and in fact this simply laced version of the theorem (Corollary 3.3) has a uniform proof, given in Sections 3-4. The remainder of the classification is proved, type-by-type in the classification of finite Coxeter groups, in Sections 5, 6, and 7. Remark 1.9. It is disappointing that some of these proofs are not uniform. However, as described above, the classification of surjective homomorphisms between weak orders on finite Coxeter groups is itself not uniform. While nice things happen quite generally, the exceptions in some types suggest that uniform arguments probably don't exist outside of the simply-laced case (Corollary 3.3). In particular, although Theorem 7.2 is a uniform statement about the crystallographic case, there is no indication that the combinatorial lattice theory of the weak order detects the crystallographic case, so a uniform proof would be surprising indeed.
In Section 8, we use the classification of surjective lattice homomorphisms between weak orders to classify surjective lattice homomorphisms between Cambrian lattices. Cambrian lattices are quotients of the weak order modulo certain congruences called Cambrian congruences. A Cambrian lattice can also be realized as a sublattice of the weak order consisting of sortable elements [24,25,30]. The significance of the Cambrian lattices begins with a collection of results, conjectured in [23] and proved in [10,24,25,30], which say that the Cambrian lattices and the related Cambrian fans encode the combinatorics and geometry of generalized associahedra of [6], which in turn provide a combinatorial model [6,7,30,31] for cluster algebras of finite type.
The classification of surjective lattice homomorphisms between Cambrian lattices, given in Theorems 8.8, 8.10, and 8.11, parallels the classification of surjective lattice homomorphisms between weak orders. The main difference is that the Cambrian lattice results have uniform statements in terms of oriented diagram homomorphisms. (However, our proofs rely on the non-uniform proofs given earlier for the weak order.) An example of a compressive homomorphism between Cambrian lattices appears as Example 8.2, which continues Example 1.1.
Interesting geometric consequences are obtained by combining the results of this paper with [22,Theorem 1.1]. The latter theorem states that every lattice congruence Θ on the weak order on W defines a polyhedral fan F Θ that coarsens the fan F (W ) defined by the reflecting hyperplanes of W . The theorem also describes the interaction between the combinatorics/geometry of the fan F Θ and the combinatorics of the quotient lattice. (In type A, the fan F Θ 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 F Θ coarsening F (W ) such that F Θ 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.
Working along the same lines for surjective congruences between Cambrian lattices, we obtain refinement relationships between Cambrian fans (fans associated to Cambrian congruences). We conclude the introduction by describing these refinement relationships in terms of dominance relationships between Cartan matrices.
ij | for all i and j. The dominance relation on Cartan matrices implies that m(r, s) ≥ m ′ (r, s) for all r, s ∈ S in the corresponding Weyl groups. In the following proposition, Φ(A) is the full root system for A, including any imaginary roots. In this paper, we only use the proposition in finite type, where there are no imaginary roots. The proposition follows from known facts about Kac-Moody Lie algebras (see Section 9), and has also been pointed out as [16,Lemma 3.5].
Proposition 1.10 may appear to be obviously false to someone who is familiar with root systems. To clarify, we emphasize that defining both Φ(A) and Φ(A ′ ) with respect to the same simple roots means identifying the root space of Φ(A) with the root space of Φ(A ′ ) by identifying the bases {α i }. Thus we may restate the proposition as follows: The set of simple root coordinate vectors of roots in Φ(A ′ ) is a subset of the set of simple root coordinate vectors of roots in Φ(A).
The refinement result on Cambrian lattices is best expressed in terms of co-roots, so we rephrase Proposition 1.10 as Proposition 9.1, which asserts a containment relation among dual root systems Φ ∨ (A) and Φ ∨ (A ′ ) when we identify the simple co-roots instead of the simple roots. The following result is proved by constructing (in Theorem 9.2) the appropriate homomorphism from W to W ′ and using it to construct a homomorphism of Cambrian lattices. Theorem 1.11. Suppose A and A ′ are Cartan matrices such that A dominates A ′ and suppose W and W ′ are the associated groups, both generated by the same set S. Suppose c and c ′ are Coxeter elements of W and W ′ respectively that can be written as a product of the elements of S in the same order. Choose a root system Φ(A) and a root system Φ(A ′ ) so that the simple co-roots are the same for the two root systems. Construct the Cambrian fan for (A, c) by coarsening the fan determined by the Coxeter arrangement for Φ(A) and construct the Cambrian fan for (A ′ , c ′ ) by coarsening the fan determined by the Coxeter arrangement for Φ(A ′ ). Then the Cambrian fan for (A, c) refines the Cambrian fan for (A ′ , c ′ ). Whereas the codimension-1 faces of the Cambrian fan for (A, c) are orthogonal to co-roots (i.e. elements of Φ ∨ (A)), the Cambrian fan for (A ′ , c ′ ) is obtained by removing all codimension-1 faces orthogonal to elements of Φ ∨ (A) \ Φ ∨ (A ′ ).
As mentioned above and as explained in [31,Section 5], Cambrian fans provide a combinatorial model for cluster algebras of finite type. The cluster-algebraic consequences of Theorem 1.11 are considered in [29]. Indeed, inspired by Theorem 1.11, the paper [29] studies much more general cluster-algebraic phenomena related to dominance relations among matrices. Remark 1.12. To the author's knowledge, the first appearance in the literature of a nontrivial surjective lattice homomorphism between finite Coxeter groups is a map found in Rodica Simion's paper [32]. (See Section 6.1.) Simion's motivations were not lattice-theoretic, so she did not show that the map is a surjective lattice homomorphism. However, 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 B n modulo the fibers of the map is isomorphic to S n+1 . It was Simion's map that first alerted the author to the fact that interesting homomorphisms exist.

Deleting vertices
In this section, we develop the most basic theory of surjective lattice homomorphisms between weak orders, leading to the proof of Theorem 1.5, which factors a surjective homomorphism into a parabolic homomorphism and a compressive homomorphism.
We assume the standard background about Coxeter groups and the weak order, which is found, for example, in [3]. (For an exposition tailored to the point of view of this paper, see [28].) As we go, we introduce background on the combinatorics of homomorphisms and congruences of finite lattices. Proofs of assertions not proved here are found in [27,.
Let (W, S) be a Coxeter system with identity element 1. The usual length function on W is written ℓ. The pairwise orders of elements r, s ∈ S are written m(r, s).
The symbol W will denote not only the group W but also a partial order, the (right) weak order on the Coxeter group W . This is the partial order on W whose cover relations are of the form w ⋖ ws for all w ∈ W and s ∈ S with ℓ(w) < ℓ(ws). The set T of reflections of W is {wsw −1 : w ∈ W, s ∈ S}. The inversion set of and element w ∈ W is inv(w) = {t ∈ T : ℓ(tw) < ℓ(w)}. An element is uniquely determined by its inversion set. The weak order on W corresponds to containment order on inversion sets. The minimal element of W is 1 and the maximal element is w 0 . We have w 0 = S.
Given J ⊆ S, the standard parabolic subgroup generated by J is written W J . This is, in particular, a lower interval in W . The maximal element of W J is w 0 (J), which equals J. We need a second Coxeter system (W ′ , S ′ ), and we use the same notation for W ′ as for W , with primes added to distinguish the groups. As a first step, we prove the following basic facts: Suppose s ∈ S has η(s) ∈ S ′ ∪ {1 ′ }. Then there exists s ′ ∈ S ′ such that η(s) > s ′ . Since η is surjective, there exists w ∈ W such that η(w) = s ′ . But η is order-preserving and s ≤ (s ∨ w), so η(s) ≤ η(s ∨ w) = η(s) ∨ s ′ = s ′ , and this contradiction shows that η(S) ⊆ S ′ ∪ {1 ′ }. We have η(S) = η(w 0 ) = w ′ 0 . But J ′ < w ′ 0 for any proper subset J ′ of S ′ , so S ′ ⊆ η(S), and we have proved (2). To prove (3), let r and s be distinct elements of S with η(r) = η(s). Then Applying η to w 0 (J) yields η( J), which equals s∈J η(s) We have proved (4). If η(r) = 1 ′ and η(s) = 1 ′ , then (3) says that η(r) = η(s) and (4) says that η restricts to a surjective homomorphism from the rank-two standard parabolic subgroup W {r,s} to the rank-two standard parabolic subgroup W ′ {η(r),η(s)} . Thus |W {r,s} | ≥ |W ′ {η(r),η(s)} |. This is equivalent to (5). Given J ⊆ S, for any w ∈ W , there is a unique factorization w = w J · J w that maximizes ℓ(w J ) subject to the constraints ℓ(w J ) + ℓ( J w) = ℓ(w) and w J ∈ W J . The element w J is also the unique element of W (and the unique element of W J ) whose inversion set is inv(w) ∩ W J . Let η J : W → W J be the map sending w to w J . We call η J a parabolic homomorphism. We now illustrate parabolic homomorphisms in terms of the usual combinatorial representations for types A n and B n and in terms of the usual geometric representation for type H 3 .
Example 2.2. We realize a Coxeter group of type A n in the usual way as the symmetric group S n+1 of permutations of {1, . . . , n + 1}, with simple generators S = {s i : i = 1, . . . n}, where each s i is the adjacent transposition (i i + 1). We write permutations π in one-line notation π = π 1 π 2 · · · π n+1 with each π i standing for π(i). Choose some k ∈ {1, . . . , n} and let J = {s 1 , . . . , s k−1 } ∪ {s k+1 , . . . , s n }. The map η J corresponds to deleting the vertex s k from the Coxeter diagram for A n , thus splitting it into components, one of type A k−1 and one of type A n−k . The map η J takes π to (σ, τ ) ∈ S k × S n+1−k , where σ is the permutation of {1, 2, . . . , k} given by deleting from the sequence π 1 π 2 · · · π n+1 all values greater than k and τ is the permutation of {1, 2, . . . , n + 1 − k} given by deleting from π 1 π 2 · · · π n+1 all values less than k + 1 and subtracting k from each value. For example, if n = 7 and k = 3, then η J (58371426) = (312, 25413). The map η J is similarly described for more general J. Example 2.3. As usual, we realize a Coxeter group of type B n as the group of signed permutations. These are permutations π of {±1, ±2, . . . , ±n} with π(−i) = −π(i) for all i. The simple generators are the permutations s 0 = (1 −1) and s i = (−i − 1 − i)(i i + 1) for i = 1, . . . , n − 1. A signed permutation π is determined by its one-line notation π = π 1 π 2 · · · π n , where each π i again stands for π(i). For J = {s 0 , . . . , s k−1 } ∪ {s k+1 , . . . , s n−1 }, the map η J corresponds to deleting the vertex s k , splitting the diagram of B n into components, of types B k and A n−k−1 . The map η J takes π to (σ, τ ), where σ is the signed permutation whose one-line notation is the restriction of the sequence π 1 π 2 · · · π n to values with absolute value less than k + 1 and τ is the permutation given by restricting the sequence (−π n )(−π n−1 ) · · · (−π 1 )π 1 · · · π n−1 π n to positive values greater than k,  Example 2.4. For W of type H 3 , we give a geometric, rather than combinatorial, description of parabolic homomorphisms. Figure 4.a shows the reflecting planes of a reflection representation of W . Each plane is represented by its intersection with a unit sphere about the origin. The sphere is considered to be opaque, so that we only see the side of the sphere that is closest to us. The spherical triangles traced out on the sphere (including those on the back of the sphere that we can't see) are in bijection with the elements of W . Specifically, the triangle corresponding to 1 is marked, and each w ∈ W corresponds to the image of the triangle marked 1 under the action of w. Taking S = {r, s, t} with m(r, s) = 5, m(s, t) = 3 and m(r, t) = 2, the reflecting planes of the reflections S are the three planes that bound the triangle marked 1. The plane corresponding to s is nearly horizontal in the picture and the plane for r intersects the plane for s at the left of the triangle marked 1. The parabolic congruence that deletes the vertex r can be seen geometrically in Figure 4.b, which shows the reflecting planes for the standard parabolic subgroup W {s,t} . The sectors cut out by these planes correspond to the elements of W {s,t} , as shown in the picture. The parabolic congruence maps an element w ∈ W , corresponding to a triangle T , to the element of The following theorem is a concatenation of [21, Proposition 6.3] and [21, Corollary 6.10]. The fact that η J is a lattice homomorphism was also established in [11].
Theorem 2.5. If J ⊆ S, then η J is a surjective homomorphism. Its fibers constitute the finest lattice congruence on W with 1 ≡ s for all s ∈ S \ J. Proposition 2.1 and Theorem 2.5 lead to the proof of Theorem 1.5. To give the proof, we need the following basic observation about a congruence Θ on a finite lattice L: Congruence classes are intervals, and the quotient L/Θ is isomorphic to the subposet of L induced by the elements of L that are the bottom elements of congruence classes. Let Θ J be the congruence whose classes are the fibers of η J . An element w ∈ W is at the bottom of its Θ J -class if and only if w ∈ W J . Thus W/Θ J is isomorphic to W J .
Proof of Theorem 1.5. If Θ is the lattice congruence on W whose congruence classes are the fibers of η, then Proposition 2.1(1) says that 1 ≡ s for all s ∈ S \ J. Theorem 2.5 says that the congruence Θ J , determined by the fibers of the homomorphism η J , is a refinement of the congruence Θ. Thus η factors as η = η ′ • ν, where ν : W → W/Θ J is the natural map. The map η ′ maps a Θ J -class to η(w) where w is any element of the Θ J -class. However, each Θ J -class contains a unique element of W J , so we can replace ν with the map η J and replace η ′ with the restriction of η to W J . By Proposition 2.1(2-3), η| WJ restricts to a bijection from J to S ′ . Theorem 1.5 reduces the problem of classifying surjective homomorphisms η between weak orders to the special case of classifying compressive homomorphisms between weak orders. As in the introduction, we may as well restrict to the case where S = S ′ and η restricts to the identity on S. For convenience, we rewrite some of the assertions of Proposition 2.1 in the case where η is compressive: Proposition 2.6. Let (W, S) and (W ′ , S) be finite Coxeter systems. If η : W → W ′ is a compressive homomorphism fixing S pointwise, then (1) If J ⊆ S, then η restricts to a surjective homomorphism W J → W ′ J . (2) m ′ (r, s) ≤ m(r, s) for each pair r, s ∈ S. Theorem 1.5 shows that surjective homomorphisms correspond to deleting vertices of the diagram and then applying a compressive homomorphism, and the second assertion of Proposition 2.6 is an additional step towards Theorem 1.2: It shows that the compressive homomorphism decreases edge labels and/or erases edges. The remainder of the paper is devoted to classifying compressive homomorphisms between finite Coxeter groups.

Erasing edges
In this section, we begin the classification of compressive homomorphisms by considering the simplest case, the case of compressive homomorphisms that erase edges but otherwise do not decrease edge labels. That is, we consider the case where, for each r, s ∈ S, either m ′ (r, s) = m(r, s) or m ′ (r, s) = 2.
Recall that the diagram of a finite Coxeter group is a forest (a graph without cycles), so removing any edge breaks a connected component of the diagram into two pieces. Given a set E of edges of the diagram for W , write S as a disjoint union of sets J 1 , J 2 , . . . , J k such that each set J i is the vertex set of a connected component of the graph obtained by deleting the edges E from the diagram. Define η E to be the map from W to W J1 ×W J2 ×· · ·×W J k that sends w ∈ W to (w J1 , w J2 , . . . , w J k ). We call η E an edge-erasing homomorphism. Beginning in this section and finishing in Section 4, we prove the following theorem.
. . , J k be the vertex sets of the connected components of the graph obtained by deleting the edges E from the diagram for W . In particular, The proof of Theorem 3.1 is similar to the proof of Theorem 1.5. We characterize the congruence associated to η E as the finest congruence containing certain equivalences and conclude that any homomorphism that erases the edges E factors through η E .
Let r and s be distinct elements of S and let m = m(r, s). Suppose r and s form an edge in E, or in other words suppose m ≥ 3. Then the standard parabolic subgroup W {r,s} is the lower interval [1, w 0 ({r, s})], consisting of two chains: 1 ⋖ r ⋖ rs ⋖ rsr ⋖ · · · ⋖ w 0 ({r, s}) and 1 ⋖ s ⋖ sr ⋖ srs ⋖ · · · ⋖ w 0 ({r, s}). We define alt k (r, s) to be the word with k letters, starting with r and then alternating s, r, s, etc. Thus the two elements covered by w 0 ({r, s}) are alt m−1 (r, s) and alt m−1 (s, r).
The key to the proof of Theorem 3.1 is the following theorem. Each map η Ji is a lattice homomorphism by Theorem 2.5, and we easily conclude that η E is a homomorphism as well. To see that η E is surjective, consider (w 1 , . . . , w k ) ∈ W J1 × W J2 × · · · × W J k . Then each w i is also an element of W . Furthermore, η Ji (w j ) = 1 if j = i. Thus the fact that η E is a lattice homomorphism implies that η E ( k i=1 w i ) = (w 1 , . . . , w k ). We have established the first assertion of Theorem 3.2. The second assertion requires more background on lattice congruences of the weak order. This background and the proof of the second assertion is given in Section 4. For now, we show how Theorem 3.2 is used to prove Theorem 3.1.
We conclude this section with some examples of edge-erasing homomorphisms.
Example 3.5. The description for type B n is similar. If E = {s k−1 -s k }, the edgeerasing homomorphism η E maps a signed permutation π to (σ, τ ) ∈ B k × S n−k+1 , where σ is the restriction of the sequence π 1 π 2 · · · π n+1 to entries π i with |π i | ≤ k and τ is obtained by restricting (−π n )(−π n−1 ) · · · (−π 1 )π 1 · · · π n−1 π n to values ≥ k and then subtracting k − 1 from each value. For example, if n = 8, k = 4, and Example 3.6. We describe edge-erasing homomorphisms from W of type H 3 in the geometric context introduced in Example 2.4. Figure 5.a-b represent the two edgeerasing homomorphisms. Thus the homomorphism erasing r-s maps each element w ∈ W , corresponding to a triangle T , to the element of W {r} × W {s,t} labeling the region in Figure 5.a containing T . Figure 5.b represents the homomorphism erasing s-t similarly. In each picture, some labels are omitted or belong on the invisible side of the sphere.

Lattice congruences of the weak order
In this section, we quote results that give us the tools to prove Theorem 3.2 and to complete the classification of compressive homomorphisms between finite Coxeter groups. We prove Theorem 3.2 at the end of this section and complete the classification in later sections. For any results that are stated here without proof or citation, proofs can be found in [21] and/or [27,.
We begin with more details on congruences on a finite lattice L. Recall from Section 1 that a congruence Θ on L is uniquely determined by the set of edges contracted by Θ (the set of edges x ⋖ y such that x ≡ y modulo Θ). In fact, Θ is uniquely determined by a smaller amount of information. An element j of L is join-irreducible if it covers exactly one element j * . We say that Θ contracts the join-irreducible element j if j ≡ j * modulo Θ. The congruence Θ is determined by the set of join-irreducible elements that Θ contracts.
The set Con(L) of all congruences on L is a sublattice of the lattice of set partitions of L. In fact, Con(L) is a distributive lattice. We write Irr(Con(L)) for the set of join-irreducible congruences on Con(L). By the Fundamental Theorem of Finite Distributive Lattices (see e.g. [33, Theorem 3.4.1]), Con(L) is isomorphic to inclusion order on the set of order ideals in Irr(Con(L)).
The weak order on a finite Coxeter group W has a special property called congruence uniformity , or sometimes called boundedness. (This was first proved in [4,Theorem 6]. See also [20,Theorem 27].) The definition of congruence uniformity is not necessary here, but congruence uniformity means in particular that the joinirreducible congruences on W (i.e. the join-irreducible elements of Con(L)) are in bijection with the join-irreducible elements of W itself. The join-irreducible elements of W are the elements j such that there exists a unique s ∈ S with ℓ(js) < ℓ(j). For each join-irreducible element j of W , the corresponding join-irreducible element Cg(j) of Con(W ) is the unique finest congruence that contracts j. Thus Irr(Con(W )) can be thought of as a partial order on the join-irreducible elements of W . Congruences on W correspond to order ideals in Irr(Con(W )). Given a congruence Θ with corresponding order ideal I, the poset Irr(Con(W/Θ)) is isomorphic to the induced subposet of Irr(Con(W )) obtained by deleting the elements of I.
The support of an element w of W is the unique smallest subset J of S such that w is in the standard parabolic subgroup W J . Given a join-irreducible element j ∈ W , the degree of j is the size of the support of j. Given a set {j 1 , . . . , j k } of join-irreducible elements of W , there exists a unique finest congruence contracting all join-irreducible elements in the set. This congruence is the join Cg(j 1 ) ∨ · · · ∨ Cg(j k ) in the congruence lattice Con(W ), or equivalently, the congruence that contracts a join-irreducible element j if and only if j is in the ideal in Irr(Con(W )) generated by {j 1 , . . . , j k }. We call this the congruence generated by {j 1 , . . . , j k }. A congruence on W is homogeneous of degree d if it is generated by a set of joinirreducible elements of degree d. Abusing terminology slightly, we call a surjective homomorphism homogeneous of degree d if its corresponding congruence is.
We restate Theorems 2.5 and 3.2 with this new terminology. (Recall that Theorem 2.5 was proven in [21]. We will prove Theorem 3.2 below.)   Lattice congruences on the weak order on a finite Coxeter group W are closely tied to the geometry of the reflection representation of W . Let Φ be a root system associated to W with simple reflections Π. For each simple reflection s ∈ S, let α s be the associated simple root. For each reflection t ∈ T , let β t be the positive root associated to t and let H t be the reflecting hyperplane for t. A point x is below H t if the inner product of x with β t is nonnegative. A set of points is below H t if each of the points is. Points and sets are above H t if the inner products are nonpositive.
The set A = {H t : t ∈ T } is the Coxeter arrangement associated to W . The arrangement A cuts space into regions, which are in bijection with the elements of W . Specifically, the identity element of W corresponds to the region D that is below every hyperplane of A, and an element w corresponds to the region wD.
A subset A ′ of A is called a rank-two subarrangement if |A ′ | > 1 and if there is some codimension-2 subspace U such that A ′ = {H ∈ A : H ⊃ U }. The subarrangement A ′ cuts space into 2|A ′ | regions. Exactly one of these regions, D ′ is below all of the hyperplanes in A ′ , and two hyperplanes in A ′ are facet-defining hyperplanes of D ′ . These two hyperplanes are called the basic hyperplanes of A ′ .
We define a cutting relation on the hyperplanes of A as follows: Given distinct hyperplanes H, H ′ ∈ A, let A ′ be the rank-two subarrangement containing H and H ′ . Then H cuts H ′ if H is a basic hyperplane of A ′ and H ′ is not a basic hyperplane of A ′ . For each H ∈ A, remove from H all points contained in hyperplanes of A that cut H. The remaining set of points may be disconnected; the closures of the connected components are called the shards in H. The set of shards of A is the union, over hyperplanes H ∈ A, of the set of shards in H. For each shard Σ, we write H(Σ) for the hyperplane containing Σ. Example 4.3. When W is a dihedral Coxeter group, the only rank-two subarrangement of A is A itself. Figure 6.a shows the reflection representation of a Coxeter group of type I 2 (5) with S = {r, s}. Figure 6.b shows the associated shards. Each of the shards contains the origin, but to make the picture legible, those shards that don't continue through the origin are drawn with an offset from the origin.  Figure 7 depicts the shards in the Coxeter arrangement of type B 3 . Here, the arrangement A is a collection of nine planes in R 3 . The shards are twodimensional cones contained in these planes. To capture the picture in the plane, we consider the intersection of A with a sphere at the origin. This intersection is an arrangement on nine great circles of the sphere. Each shard, intersected with the sphere, is either an entire great circle or an arc of a great circle. The figure shows these intersections under a stereographic projection from the sphere to the plane. The region D is the small triangle that is inside all of the circles. As in Figure 6, where shards intersect, those shards that do not continue through the intersection are shown with an offset from the intersection. Two of the shards are distinguished by arrows. The significance of these two shards will be explained in Example 4.5.
The shards of A are in one-to-one correspondence with the join-irreducible elements of W . For each shard Σ, an upper element of Σ is an element w ∈ W such that the region wD is above H(Σ) and intersects Σ in codimension 1. The set U (Σ) of upper elements of Σ contains exactly one element j(Σ) that is join-irreducible in W . The element j(Σ) is the unique minimal element of U (Σ) in the weak order. This is a bijection from shards to join-irreducible elements. The inverse map sends a join-irreducible element j to Σ(j), the shard that contains jD ∩ (j * D).
Shards in A correspond to certain collections of edges in the Hasse diagram of the weak order on W . Specifically, a shard Σ corresponds to the set of all edges x ⋖ y such that (xD ∩ yD) ⊆ Σ. For each shard Σ, a congruence Θ either contracts none of the edges associated to Σ or contracts all of the edges associated to Σ. (See [26,Proposition 6.6].) If Θ contracts all of the edges associated to Σ, then we say that Θ removes Σ. In particular, Θ removes a shard Σ if and only if it contracts the join-irreducible element j(Σ). For any congruence Θ, the set of shards not We now define a directed graph on shards, called the shard digraph. Given two shards Σ and Σ ′ , say Σ → Σ ′ if H(Σ) cuts H(Σ ′ ) and Σ∩Σ ′ has codimension 2. The digraph thus defined on shards is called the shard digraph. This digraph is acyclic 2 and we call its transitive closure the shard poset. The bijection Σ → j(Σ) from shards to join-irreducible elements is an isomorphism from the shard poset to the poset Irr(Con(W )), thought of as a partial order on join-irreducible elements of W . Thus, given any congruence Θ, the set of shards removed by Θ is an order ideal in the shard poset. Conversely, for any set of shards forming an order ideal in the shard poset, there is a congruence removing exactly that set of shards.
We use this correspondence to reuse shard terminology for join-irreducible elements and vice versa. So, for example, we talk about the degree of a shard (meaning the degree of the corresponding join-irreducible element), etc.
Example 4.5. Recall that Example 1.1 started by choosing a pair of edges in the weak order on a Coxeter group of type B 3 . The finest congruence contracting the two edges was calculated, and the quotient modulo this congruence was found to be isomorphic to the weak order on a Coxeter group of type A 3 . The characterization of congruences in terms of shards allows us to revisit this example from a geometric point of view. Contracting the two chosen edges corresponds to removing the two shards indicated with arrows in Figure 7. Removing these two shards forces 2 Shards are often considered in a more general context of simplicial hyperplane arrangements.
In this broader setting, the shard digraph need not be acyclic. See [20, Figure 5]. the removal of all shards below them in the shard poset. Figure 8 depicts the shards whose removal is not forced. (Gaps between intersecting shards have been closed in this illustration.) The resulting fan is piecewise-linearly (but not linearly) equivalent to the fan defined by a Coxeter arrangement of type A 3 .
Let α denote the involution w → ww 0 on W. This is an anti-automorphism of the weak order. For each congruence Θ on W, let α(Θ) be the antipodal congruence to Θ, defined by x ≡ y mod α(Θ) if and only if α(x) ≡ α(y) mod Θ. The involution α induces an anti-isomorphism from W/Θ to W/(α(Θ)).
The following is a restatement of part of [21, Proposition 6.13].
Proposition 4.6. Let Θ be a lattice congruence on W , let r, s ∈ S, and let The following lemma is part of [26,Lemma 3.11], rephrased in the special case of the weak order. (Cf. [21,Lemma 3.9].) The following lemma is a rephrasing of [26,Lemma 3.12] in the special case of the weak order. (Cf. [25,Lemma 4.6].) It implies in particular that a shard of degree 2 is only arrowed by shards of degree 1.
Lemma 4.8. Let Σ be a shard contained in a reflecting hyperplane H t . The Σ has exactly one facet if and only if t ∈ S but t ∈ W {r,s} for some distinct r, s ∈ S. In that case, the unique facet of Σ is H r ∩ H s .
We now use the machinery of shards to prove Theorem 3.2.

Proof of Theorem 3.2.
Recall that the first assertion of the theorem has already been established. As before, let Θ E be the lattice congruence whose classes are the fibers of η E . It remains only to prove that Θ E is generated by the join-irreducible elements alt k (r, s) and alt k (s, r) for all {r, s} ∈ E and k = 2, 3, . . . , m(r, s) − 1.
Let s ∈ S and w ∈ W have ws ⋖ w and let t ∈ T be the reflection wsw −1 , so that ws = tw. The inversion set of w Ji is inv(w) ∩ W Ji and the inversion set of an element determines the element, so (ws) Ji = w Ji if and only if t ∈ W Ji . Thus the edge ws ⋖ w is contracted by Θ E if and only if wsw −1 ∈ W Ji for all i ∈ {1, . . . , k}. Equivalently, a shard is removed by Θ E if and only if the reflecting hyperplane containing it is H t for some reflection t with t ∈ W Ji for all i ∈ {1, . . . , k}.
Let Σ k (r, s) be the shard associated to the join-irreducible element alt k (r, s) for each k ∈ {2, . . . , m(r, s) − 1}. To prove the theorem, we will show that every shard removed by Θ E is forced by the removal of all shards of the form Σ k (r, s) and Σ k (s, r) for each edge r-s in E. Specifically, let Σ be a shard removed by Θ E . We complete the proof by establishing the following claim: If Σ is not Σ k (r, s) or Σ k (s, r) for some edge r-s in E and some k ∈ {2, . . . , m(r, s) − 1}, then there exists another shard Σ ′ , also removed by Θ E , such that Σ ′ → Σ in the shard digraph. Since the shard digraph is acyclic, the claim will imply that, for every shard Σ removed by Θ E , there is a directed path from a shard Σ k (r, s) or Σ k (s, r) to Σ.
We now prove the claim. Since Σ is removed by Θ E , it is contained in a reflecting hyperplane H t with t ∈ W Ji for all i ∈ {1, . . . , k}. Consider a reflecting hyperplane H t ′ that cuts H t to define a facet of Σ. Then H t ′ is basic in the rank-two subarrangement A ′ containing H t ′ and H t , while H t is not basic in A ′ . The other basic hyperplane, H t ′′ , of A ′ also cuts H t to define the same facet of Σ. Thus there exists a shard Σ ′ contained in H t ′ such that Σ ′ → Σ, and there exists a shard Σ ′′ contained in H t ′′ such that Σ ′′ → Σ. If The positive root β t is in the positive linear span of β t ′ and β t ′′ . But the simple root coordinates (coordinates with respect to the basis Π) of β t ′ and β t ′′ are supported on the disjoint subsets {α s : s ∈ J i } and {α s : s ∈ J j }. Thus the simple root coordinates of β t ′ are determined, up to scaling, by the simple root coordinates of β t , by restricting the coordinates of β t to simple roots in {α s : s ∈ J i }. The simple root coordinates of β t ′′ are determined similarly, up to scaling, by the simple root coordinates of β t .
We conclude that there is at most one facet of Σ defined by hyperplanes H t ′ and H t ′′ such that there exists i ∈ {1, . . . , k} with t ′ ∈ W Ji and j ∈ {1, . . . , k} with t ′′ ∈ W Jj . Thus if Σ has more than one facet, then we can use one of these facets to find, as above, a shard Σ ′ , removed by Θ E , with Σ ′ → Σ. If Σ has only one facet, then Lemma 4.8 says that the facet of Σ is defined by H r and H s for some r, s ∈ S. In this case, the join-irreducible element j(Σ) associated to Σ is in W {r,s} but not in {r, s}. Thus j(Σ) is alt k (r, s) and alt k (s, r) for some k ∈ {2, 3, . . . , m(r, s) − 1}. Equivalently Σ = Σ k (r, s) or Σ = Σ k (s, r) for some k ∈ {2, 3, . . . , m(r, s) − 1}. Since Σ is removed by Θ E , the edge r-s is in E. Since j(Σ) ∈ {r, s}, in particular t ∈ S, so Lemma 4.7 rules out the possibility that Σ has no facets. We have proved the claim, and thus completed the proof of the theorem.
We conclude the section by describing the shard digraph for types A n and B n , quoting results of [21,. The descriptions use the combinatorial realizations explained in Examples 2.2 and 2.3.
For any subset A of {1, . . . , n + 1}, let A c = [n + 1] \ A, let m be the minimum element of A and let M be the maximum element of A c . Join-irreducible elements of S n+1 correspond to nonempty subsets of [n + 1] such that M > m. Given such a subset, the corresponding join-irreducible permutation γ is constructed by listing the elements of A c in ascending order followed by the elements of A in ascending order. The poset Irr(Con(S n+1 )) can be described combinatorially, as a partial order on join-irreducible elements, in terms of these subsets. (See [21,Theorem 8.1].) In this paper, we do not need the details, except for the case n = 3, which is shown in Figure 9.
We do, however, need details of the description of Irr(Con(B n )). A signed subset A is a subset of {±1, ±2, . . . , ±n} such that A contains no pairs {−i, i}. such a signed subset A, the corresponding join-irreducible permutation γ has oneline notation given by the elements of {1, . . . , n} \ {|a| : a ∈ A} in ascending order, followed by the elements of A in ascending order. The reflection t associated to the unique cover relation down from γ is The associated shard is in the hyperplane H t . Figure 10 shows the poset Irr (Con(B 3 )). To describe this poset for general n, we quote the combinatorial description of the shard digraph in type B n . Arrows between shards depend on certain combinations of conditions. The meaning of the letters q, f and r in the labels for the conditions is explained in [21,Section 7]. We begin with conditions (q1) through (q6). ( Next we define condition (f), which depends on a parameter in {±1, . . . , ±n}. In the following conditions, the superscript "c" mean complementation in the set

Decreasing edge labels
Theorem 3.1 reduces the problem of classifying compressive homomorphisms η between weak orders to the special case of compressive homomorphisms that do not erase any edges. Equivalently, these are the compressive homomorphisms between finite Coxeter groups that restrict to (unlabeled) graph isomorphisms of diagrams. As a byproduct, Theorem 3.1 allows us to further reduce the problem to the case where the diagrams are connected (or equivalently, where the Coxeter groups are irreducible). We continue to take S to be the set of simple generators of W and of W ′ and assume that η : W → W ′ is such a homomorphism that fixes S pointwise, so that the diagrams of W and W ′ are identical as unlabeled graphs. Recall that Proposition 2.6 says that for each edge in the diagram of W , the corresponding edge in the diagram of W ′ has weakly smaller label.
Up to now, the classification of surjective lattice homomorphisms between weak orders has proceeded by uniform arguments, rather than arguments that are specific to particular families in the classification of finite Coxeter groups. To complete the classification of homomorphisms, we now turn to the classification of finite Coxeter groups, which tells us in particular that there are very few cases remaining to consider. We need to study the cases where (W, W ′ ) are (I 2 (m), The cases where (W, W ′ ) are (I 2 (m), I 2 (m ′ )) for m ′ < m are easily understood by considering Example 4.3, where W is I 2 (5). It is apparent from Figure 6 that a congruence Θ has the property that W/Θ is isomorphic to the weak order on I 2 (4) if and only if Θ contracts exactly one of the join-irreducible elements rs, rsr, and rsrs and exactly one of the join-irreducible elements sr, srs, srsr.
More generally, let W be a Coxeter group of type I 2 (m) with S = {r, s}. A congruence Θ on W has the property that W/Θ is isomorphic to the weak order on I 2 (m ′ ) if and only if Θ contracts exactly m − m ′ join-irreducible elements of the form alt k (r, s) for k ∈ {2, . . . , m − 1} and exactly m − m ′ join-irreducible elements of the form alt k (s, r) for k ∈ {2, . . . , m − 1}. This result for dihedral groups is simple, but its importance for the general case is underscored by Theorem 1.7.
In Sections 6 and 7, we consider the remaining possibilities for the pair (W, W ′ ) and, in particular, complete the proofs of Theorems 1.6 and 1.7. As mentioned earlier, Theorem 1.2 follows from Theorems 1.5 and 1.6.
6. Homomorphisms from B n to S n+1 We realize the groups S n+1 and B n as in Examples 2.2 and 2.3. A surjective homomorphism η from B n to S n+1 restricts, by Proposition 2.6, to a surjective homomorphism from (B n ) {s0,s1} ∼ = B 2 to (S n+1 ) {s1,s2} ∼ = S 3 . Thus, as discussed in Section 5, the congruence associated to η contracts exactly one of the joinirreducible elements s 0 s 1 , s 0 s 1 s 0 , and exactly one of the join-irreducible elements s 1 s 0 , s 1 s 0 s 1 . We will see that each of the four choices leads to a unique surjective homomorphism and that three of the four choices are associated to homogeneous congruences of degree 2.
6.1. Simion's homomorphism. We consider first the case where s 0 s 1 and s 1 s 0 s 1 are contracted, but we start not by contracting join-irreducibles, but by giving a map from B n to S n+1 . This map was first defined by Simion [32] in connection with a construction of the type-B associahedron (also known as the cyclohedron).
Suppose π ⋖ τ in the weak order on B n . Then the one-line notations for π and τ differ in one of two ways: Either they agree except in the sign of the first entry or they agree except that two adjacent entries of π are transposed in τ . If they agree except in the sign of the first entry, then η σ (π) = η σ (τ ). If they agree except that two adjacent entries of π are transposed in τ , then η σ (π) = η σ (τ ) if and only if the two adjacent entries have opposite signs. We have proved the following: Proposition 6.2. If π ⋖ τ in the weak order on B n , then η σ (π) = η σ (τ ) if and only if the reflection t associated to the cover π ⋖ τ is (i − j)(j − i) for some i and j with 1 ≤ i < j ≤ n.
We give a similar characterization of the congruence generated by s 0 s 1 and s 1 s 0 s 1 : Proposition 6.3. The congruence generated by s 0 s 1 and s 1 s 0 s 1 removes a shard Σ if and only if Σ is contained in a hyperplane H t such that t = (i − j)(j − i) for some i and j with 1 ≤ i < j ≤ n.
Proof. Let C be the set of signed subsets corresponding to shards contained in hyperplanes H t such that t = (i − j)(j − i) for some i and j with 1 ≤ i < j ≤ n. Thus C is the set of all nonempty signed subsets A with m < 0 and m = −M . (Equivalently, A ⊆ {1, . . . , n} and |A| < n.) The assertion of the proposition is that C is the set of signed subsets representing shards removed by the congruence generated by s 0 s 1 and s 1 s 0 s 1 . The join-irreducible element s 0 s 1 has one-line notation 2(−1)34 · · · n, and the join-irreducible element s 1 s 0 s 1 is 1(−2)34 · · · n. The corresponding signed subsets are {−1, 3, 4, . . . , n} and {−2, 3, 4, . . . , n}, both of which are in C.
By definition of M 1 , the element −M 1 is not in A 1 , and since A 2 ⊂ A 1 , we conclude that −M 1 ∈ A 2 . Thus (f2 : M 1 ) holds. Also, (r1) holds, so A 1 → A 2 in the shard digraph. So far, we know nothing about how η σ relates to the weak order on S n+1 . But Propositions 6.2 and 6.3 constitute a proof of the second assertion of Theorem 6.1: that the congruence Θ σ defined by the fibers of η σ is generated by s 0 s 1 and s 1 s 0 s 1 . Proposition 6.2 implies that the bottom elements of Θ σ are exactly those signed permutations whose one-line notation consists of a (possibly empty) sequence of negative entries followed by a (possibly empty) sequence of positive entries. The restriction of η σ to bottom elements is a bijection to S n+1 , with inverse map described as follows: If τ ∈ S n+1 and τ i = 1, then the inverse map takes τ to the signed permutation whose one-line notation is This inverse map is easily seen to be order-preserving. The map η σ is also easily seen to be order-preserving, so its restriction is as well. We have shown that the restriction of η σ to bottom elements of Θ σ is an isomorphism to the weak order on S n+1 .
The natural map from B n to B n /Θ σ is a surjective homomorphism, and B n /Θ σ is isomorphic to the subposet of B n induced by bottom elements of Θ σ . Since η σ equals the composition of the natural map, followed by the isomorphism to the poset of bottom elements, followed by the isomorphism to S n+1 , it is a surjective homomorphism. Now, if η is any surjective homomorphism agreeing with η σ on (B n ) {s0,s1} , the associated congruence Θ contracts s 0 s 1 and s 1 s 0 s 1 . Thus Θ is a coarsening of the congruence Θ σ associated to η σ . But then since both B n /Θ and B n /(Θ σ ) are isomorphic to S n+1 , they must coincide. We have completed the proof of Theorem 6.1.

6.2.
A non-homogeneous homomorphism. In this section, we consider surjective homomorphisms whose congruence contracts s 0 s 1 s 0 and s 1 s 0 . We begin by defining a homomorphism η ν that is combinatorially similar to Simion's homomorphism. We will see that, whereas η σ is homogeneous of degree 2, η ν is non-homogeneous. However, η ν is still of low degree: it is generated by contracting join-irreducible elements of degrees 2 and 3.
The outline of the proof of Theorem 6.4 is the same as the proof of Theorem 6.1. We first determine when signed permutations π ⋖ τ in B n map to the same element of S n+1 . If π and τ agree except in the sign of the first entry (with π 1 necessarily positive), then η ν (π) = η ν (τ ) if and only if π 1 > 1. If they agree except that two adjacent entries of π are transposed in τ , then η ν (π) = η ν (τ ) if and only if the two transposed entries have opposite signs and neither is ±1. Thus: Proposition 6.5. If π ⋖ τ in the weak order on B n , then η ν (π) = η ν (τ ) if and only if the reflection t associated to the cover π ⋖ τ is either (i − i) for some i > 1 or (i − j)(j − i) for some i and j with 2 ≤ i < j ≤ n.
We now show that the congruence generated by s 0 s 1 s 0 , s 1 s 0 , s 1 s 0 s 1 s 2 , and s 2 s 1 s 0 s 1 s 2 agrees with the fibers of η ν , as described in Proposition 6.5. Proposition 6.6. In the congruence generated by s 0 s 1 s 0 , s 1 s 0 , s 1 s 0 s 1 s 2 , and s 2 s 1 s 0 s 1 s 2 a shard is removed if and only if it is contained in a hyperplane H t with t = (i − i) and i > 1 or t = (i − j)(j − i) and 2 ≤ i < j ≤ n.
Proof. Let C be the set of signed subsets corresponding to shards contained in hyperplanes H t such that t is either (i − i) for some i > 1 or (i − j)(j − i) for some i and j with 2 ≤ i < j ≤ n. Thus C is the set of all nonempty signed subsets We first show that no set A 1 ∈ C arrows a set A 2 ∈ C. Let A 1 ∈ C and A 2 ∈ C. Then m 1 < −1 and M 1 > 1, while either m 2 ≥ −1 or M 2 = 1. The fact that m 1 < −1 rules out condition (q2). If M 2 = 1, then the fact that M 1 > 1 rules out conditions (q1), (q3) and (q4), and the fact that m 1 < −1 rules out (q5) and (q6). Suppose m 2 ≥ −1. Then the equality M 2 = −m 2 in (q1) fails unless m 2 = −1, but in that case M 2 = 1, so (q1) is ruled out. The fact that m 1 < −1 rules out (q3) and (q4). Since −m 2 ≤ 1 and M 1 > 1, conditions (q5) and (q6) fail as well.
If τ i = 2 and τ j = 1, with i < j, then the inverse map takes τ to the same signed permutation, except with −1 in place of 1. This inverse in order-preserving, and we have proved the first two assertions of Theorem 6.4.
To prove the third assertion, we temporarily introduce the bulky notation η (n) ν : B n → S n+1 . (Until now, we had suppressed the explicit dependence of the map η ν on n, and we will continue to do so after this explanation.) Arguing as in the proof of Theorem 6.1, we easily see that, for each n ≥ 3, the map η (n) ν is the unique surjective homomorphism from B n to S n+1 whose restriction to (B n ) {s0,s1,s2} is η ν . Now let η be any surjective homomorphism from B n to S n+1 whose restriction to (B n ) {s0,s1} is η (2) ν . Let Θ be the associated congruence on B n . Then in particular, the congruence defined by the restriction of η to (B n ) {s0,s1,s2} ∼ = B 3 contracts s 0 s 1 s 0 and s 1 s 0 , corresponding to signed subsets {−2, −1, 3} and {−2, 1, 3}. Figure 11 shows the poset of signed subsets corresponding to join-irreducible elements in (B n ) {s0,s1,s2} ∼ = B 3 not forced to be contracted by the contraction of s 0 s 1 s 0 and s 1 s 0 . (This is the complement of an order ideal in the poset of Figure 10.) By Proposition 2.6, the restriction of η to (B n ) {s0,s1,s2} is a surjective homomorphism to S 4 . In particular, the restriction of Irr (Con((B n ) {s0,s1,s2} )) to joinirreducibles not contracted by Θ is isomorphic to Irr(Con(S 4 )). Comparing Figure 11 to Figure 9, we see that Θ must contract two additional join-irreducible  Figure 11. The complement of an order ideal in Irr(Con(W )) for W of type B 3 .
elements, beyond those forced by s 0 s 1 s 0 and s 1 s 0 . We see, furthermore, that the only two join-irreducible elements that can be contracted, to leave a poset isomorphic to Irr(Con(S 4 )), are those whose signed subsets are −2 and −3. We conclude that Θ contracts s 1 s 0 s 1 s 2 and s 2 s 1 s 0 s 1 s 2 . Theorem 6.4 says that Θ is weakly coarser than Θ ν . But since η and η ν are both surjective lattice homomorphisms to S n+1 , the congruences Θ and Θ ν have the same number of congruence classes, so Θ = Θ ν . Thus η and η ν agree up to automorphisms of S n+1 , but the only nontrivial automorphism of S n+1 is the diagram automorphism. Since both maps take (B n ) {s0,s1} to (S n+1 ) {s1,s2} , we rule out the diagram automorphism and conclude that η = η ν . Thus completes the proof of Theorem 6.4.
6.3. Two more homogeneous homomorphisms. In this section, we consider the case where s 0 s 1 and s 1 s 0 are contracted and the case where s 0 s 1 s 0 and s 1 s 0 s 1 are contracted. The congruences associated to these cases are dual to each other by Proposition 4.6. Let η δ : B n → S n+1 send π ∈ B n to the permutation obtained as follows: If the one-line notation for π contains the entry 1, then construct a sequence (−π n )(−π n−1 ) · · · (−π 1 ) 0 π 1 · · · π n−1 π n , extract the subsequence consisting of nonnegative entries, and add 1 to each entry. If the one-line notation for π contains the entry −1, then extract the subsequence of (−π n )(−π n−1 ) · · · (−π 1 )π 1 · · · π n−1 π n consisting of values greater than or equal to −1, change −1 to 0, then add 1 to each entry. Notice that η δ is a hybrid of η σ and η ν , in the sense that η δ (π) = η σ (π) if the one-line notation of π contains 1 and η δ (π) = η ν (π) if the one-line notation of π contains −1.
We will prove the following theorem: Theorem 6.7. The map η δ is a surjective lattice homomorphism from B n to S n+1 . Its fibers constitute the congruence generated by s 0 s 1 s 0 and s 1 s 0 s 1 . Furthermore, η δ is the unique surjective lattice homomorphism from B n to S n+1 whose restriction to (B n ) {s0,s1} agrees with η δ .
Theorem 6.7 implies in particular that the lattice homomorphism associated to the congruence of Example 1.1 is the n = 3 case of η δ .
Suppose π ⋖ τ in B n . First, suppose that 1 is an entry in the one-line notation of both π and τ . If π and τ agree except in the sign of the first entry, then η δ (π) = η δ (τ ). If they agree except that two adjacent entries of π are transposed in τ , then η δ (π) = η δ (τ ) if and only if the two adjacent entries have opposite signs. Next, suppose that −1 is an entry in the one-line notation of both π and τ . If π and τ agree except in the sign of the first entry, then since −1 is an entry in the one-line notation of both π and τ , we must have π 1 > 1, and therefore η δ (π) = η δ (τ ). If they agree except that two adjacent entries of π are transposed in τ , then η ν (π) = η ν (τ ) if and only if the two transposed entries have opposite signs and neither is −1. Finally, suppose π has the entry 1 in its one-line notation, but τ has −1. Then π 1 = 1 and τ 1 = −1, and η δ (π) = η δ (τ ). Thus: Proposition 6.8. Suppose π ⋖τ in the weak order on B n , and let t be the reflection associated to the cover π ⋖ τ . Then η δ (π) = η δ (τ ) if and only if one of the following conditions holds: (ii) π has the entry 1 in its one-line notation and t is (1 − j)(j − 1) for some j with 1 < j ≤ n. (iii) π has the entry −1 in its one-line notation and t is (i − i) for some i > 1.
As in the previous cases, we now show that the congruence generated by s 0 s 1 s 0 and s 1 s 0 s 1 has a description compatible with Proposition 6.8.
for some i > 1.
Before proving Proposition 6.9, we verify that conditions (ii) and (iii) make sense. Specifically, in both conditions, we rule out the possibility that Σ is neither above nor below H (1 −1) . Note that if 1 < j ≤ n and t = (1 −j)(j −1), then the rank-two subarrangement containing H (1 −1) and H t has basic hyperplanes H (1 −1) and H t ′ , where t ′ = (1 j)(−j −1). Thus every hyperplane H t , for t as in condition (ii), is cut at H (1 −1) , and thus every shard in H t is either above or below H (1 −1) . Similarly, every shard in H t , for t as in condition (iii), is either above or below H (1 −1) . H (1 −1) , so that Σ is either above or below H (1 −1) . Then Σ is above H (1 −1) if and only if its associated join-irreducible element γ has −1 in its one-line notation. This occurs if and only if −1 is contained in the signed subset representing γ. The shards specified by conditions (i)-(iii) in Proposition 6.9 correspond, via join-irreducible elements, to signed subsets A described respectively by the following conditions: For condition (ii), the condition is (m, M ) ∈ {(−1, j), (−j, 1)} for some 1 < j ≤ n, and −1 ∈ A. However, the requirement that −1 ∈ A implies that m = −1, so (m, M ) = (−j, 1), as indicated above.

Proof. Suppose Σ is a shard in a hyperplane that is cut by
Let C be the set of signed subsets satisfying (i), (ii) or (iii). We can describe C more succinctly as the set of signed subsets satisfying both of the following conditions: We now show that no set A 1 ∈ C arrows a set A 2 ∈ C. Let A 1 ∈ C and A 2 ∈ C. Suppose m 2 ≥ −1. Then (q1) and (q6) fail, because each would include the impossible assertion that M 1 < −m 2 . Also, (q2)-(q4) fail because m 1 < −1.
Combining Propositions 6.8 and 6.9, the fibers of η δ are a lattice congruence Θ δ on B n satisfying the second assertion of Theorem 6.7. We now show that η δ is order-preserving, that η δ restricts to a bijection from bottom elements of Θ δ -classes to permutations in S n+1 and that the inverse of the restriction is order-preserving.
To see that η δ is order-preserving, suppose π ⋖ τ in B n . If 1 is an entry in the one-line notation of both π and τ , then η δ coincides with η σ on π and τ , so η δ preserves the order relation π ≤ τ . Similarly, if −1 is an entry in the one-line notation of both π and τ , then η δ preserves the order relation π ≤ τ because η ν is order-preserving. Finally, if 1 is an entry in the one-line notation of π and −1 is an entry in the one-line notation of τ , then η δ (π) and η δ (τ ) both have 1 and 2 adjacent but in different orders. Otherwise, the two permutations agree, and we see that η δ (π) ⋖ η δ (τ ). We have verified that η δ is order-preserving. Proposition 6.8 leads to a characterization of the bottom elements of Θ δ -classes. A signed permutation π whose one-line notation contains 1 is a bottom element if and only if its one-line notation consists of a (possibly empty) sequence of negative entries followed by a sequence of positive entries (including 1). A signed permutation with −1 in its one-line notation is a bottom element if and only if it consists of a (possibly empty) sequence of positive entries, followed by a sequence of negative elements beginning with −1, and finally a (possibly empty) sequence of positive elements. Notice that the bottom elements of Θ δ that have 1 in their one-line notation map to permutations with 1 preceding 2, and the bottom elements of Θ δ that have −1 in their one-line notation map to permutations with 2 preceding 1.
It is now easily verified that the inverse is order-preserving. The third assertion of Theorem 6.7 follows as in the case of Theorem 6.1, and we have completed the proof of Theorem 6.7. Now let η ǫ : B n → S n+1 send π ∈ B n to the permutation obtained as follows: If the one-line notation for π contains the entry 1, then extract the subsequence of (−π n )(−π n−1 ) · · · (−π 1 )π 1 · · · π n−1 π n consisting of values greater than or equal to −1, change −1 to 0, then add 1 to each entry. If the one-line notation for π contains the entry −1, then construct a sequence (−π n )(−π n−1 ) · · · (−π 1 ) 0 π 1 · · · π n−1 π n , extract the subsequence consisting of nonnegative entries, and add 1 to each entry. Thus η ǫ is a hybrid of η σ and η ν in exactly the opposite way that η δ is a hybrid: η ǫ (π) = η ν (π) if the one-line notation of π contains 1 and η ǫ (π) = η σ (π) if the one-line notation of π contains −1.
Theorem 6.10. The map η ǫ is a surjective lattice homomorphism from B n to S n+1 . Its fibers constitute the congruence generated by s 0 s 1 and s 1 s 0 . Furthermore, η ǫ is the unique surjective lattice homomorphism from B n to S n+1 whose restriction to (B n ) {s0,s1} agrees with η ǫ .
The maps rev and neg both send a group element w to ww 0 , where w 0 is the longest element of the corresponding Coxeter group (the element (n + 1)n · · · 1 in S n+1 or the element (−1) · · · (−n) in B n ). The map w → ww 0 is an antiautomorphism of the weak order on any finite Coxeter group. Thus the first assertion of Theorem 6.10 follows from the first assertion of Theorem 6.7. Furthermore, by Proposition 4.6, the second assertion of Theorem 6.10 follows from the second assertion of Theorem 6.7, and the third assertion follows as usual. Homomorphisms from type F 4 . Let W be a Coxeter group of type F 4 and with S = {p, q, r, s}, m(p, q) = 3, m(q, r) = 4 and m(r, s) = 3. Figure 12 shows the order ideal in the weak order on W that encodes these values of m. This is comparable to Figure 2.a, except that the square intervals that indicate where m is 2 (e.g. m(p, r) = 2) are omitted. For W ′ of type A 4 (i.e. W ′ = S 5 ), we identify p, q, r, s with s 1 , s 2 , s 3 , s 4 (in that order) to discuss homomorphisms fixing S pointwise.

Homomorphisms from exceptional types
Theorem 7.1. There are exactly four surjective lattice homomorphisms from F 4 to A 4 that fix S pointwise: For each choice of γ 1 ∈ {qr, qrq} and γ 2 ∈ {rq, rqr}, there exists a unique such homomorphism whose associated congruence contracts γ 1 and γ 2 .
Proof. Suppose η : F 4 → A 4 be a surjective homomorphism fixing S pointwise. By Proposition 2.6 and the discussion in Section 5, the congruence Θ on F 4 defined by the fibers of η contracts exactly one of qr and qrq and exactly one of rq and rqr. In each of the four cases, we verify that there is a unique choice of η. Case 1: Θ contracts qr and rq. Computer calculations show that the quotient of W modulo the congruence generated by qr and rq is isomorphic to the weak order on A 4 . Thus a unique η exists in this case. Case 2: Θ contracts qrq and rqr. Computer calculations as in Case 1 (or combining Case 1 with Proposition 4.6) show that a unique η exists.
Case 3: Θ contracts qrq and rq. Proposition 2.6 implies that the restriction to W {q,r,s} is a surjective lattice homomorphism. Thus Theorem 6.4 implies that the restriction of η to W {q,r,s} agrees with η ν . In particular, in addition to the joinirreducible elements of W {q,r,s} contracted by the congruence generated by qrq and rq, the congruence Θ contracts rqrs and srqrs. Computer calculations show that the quotient of W modulo the congruence generated by qrq, rq, rqrs, and srqrs is isomorphic to the weak order on A 4 . Thus a unique η exists. Case 4: Θ contracts qr and rqr. By an argument/calculation analogous to Case 3 (or by Case 3 and the diagram automorphism of F 4 ), a unique η exists. The congruence Θ is generated by qr, rqr, qrqp, and pqrqp.
We have dealt with the final crystallographic case, and thus proved the following existence and uniqueness theorem. Homomorphisms from type H 3 . Now, let W be a Coxeter group of type H 3 and with S = {q, r, s}, m(q, r) = 5 and m(r, s) = 3. Identify q, r, s with the generators s 1 , s 2 , s 3 of A 3 = S 4 . Theorem 7.3. There are exactly nine surjective lattice homomorphisms from H 3 to A 3 that fix S pointwise: For each choice of distinct γ 1 , γ 2 ∈ {qr, qrq, qrqr} and distinct γ 3 , γ 4 ∈ {rq, rqr, rqrq}, there exists a unique such homomorphism whose associated congruence contracts γ 1 , γ 2 , γ 3 and γ 4 .
Proof. Suppose η : H 3 → A 3 is a surjective homomorphism, fixing S pointwise, whose fibers define a congruence Θ. As in the proof of Theorem 7.1, we consider the nine cases separately.
In each of Cases 1-6 below, computer calculations show that the quotient of H 3 modulo the congruence generated by the given join-irreducible elements is isomorphic to the weak order on A 3 . Thus in each of these cases, a unique η exists. Case 1: Θ contracts qr, qrq, rq, and rqr. Case 2: Θ contracts qr, qrq, rq, and rqrq. Case 3: Θ contracts qr, qrq, rqr, and rqrq. Case 4: Θ contracts qr, qrqr, rq, and rqrq. Case 5: Θ contracts qr, qrqr, rqr, and rqrq. Case 6: Θ contracts qrq, qrqr, rqr, and rqrq. Case 7: Θ contracts qrq, qrqr, rq, and rqr. In this case, the quotient of H 3 modulo the congruence Θ ′ generated by the given join-irreducible elements is a poset with 28 elements. Inspection of Irr(Con(H 3 /Θ ′ )) reveals a unique way to contract additional join-irreducible elements so as to obtain Irr (Con(A 3 )). (Cf. Figure 11 and the surrounding discussion in Section 6.2.) The additional join-irreducible elements to be contracted are rqrqsrq and srqrqsrq. Computer calculations confirm that the quotient of H 3 modulo the congruence generated by qrq, qrqr, rq, rqr, rqrqsrq, and srqrqsrq is indeed isomorphic to A 3 , so there exists a unique η in this case. Case 8: Θ contracts qr, qrqr, rq, and rqr. Computer calculations show that the quotient of H 3 modulo the congruence generated by qr, qrqr, rq, rqr, rqrqsrqr, and srqrqsrqr is isomorphic to A 3 . As in Case 7, inspection of the poset of irreducibles of the congruence lattice of the quotient reveals that no other possibilities exist. Thus η exists and is unique in this case. Case 9: Θ contracts qrq, qrqr, rq, and rqrq. The existence and uniqueness of η follows from Case 7 and Proposition 4.6. Alternately, the calculation can be made directly as in Cases 7-8, realizing Θ as the congruence generated by qrq, qrqr, rq, rqrq, rqrs, and srqrs.
Homomorphisms from type H 4 . Finally, let W be a Coxeter group of type H 4 and with S = {q, r, s, t}, m(q, r) = 5, m(r, s) = 3 and m(s, t) = 3. The classification of surjective homomorphisms from H 4 to A 4 and from H 4 to B 4 exactly follows the classification of surjective homomorphisms from H 3 to A 3 and from H 3 to B 3 , as we now explain.
Let η be any surjective homomorphism η from H 4 to A 4 or B 4 with associated congruence Θ. By Proposition 2.6, the restriction η ′ of η to the standard parabolic subgroup W {q,r,s} (of type H 3 ) is a homomorphism from H 3 to A 3 or B 3 . Thus η ′ is described by Theorem 7.3 or 7.4. The congruence Θ ′ associated to η ′ is the restriction of Θ to W {q,r,s} . The proofs of Theorems 7.3 and 7.4, determine, for each surjective homomorphism, a set Γ of join-irreducibles that generate the associated congruence. Since Θ ′ agrees with Θ on W {q,r,s} , the congruence associated to η must also contract the join-irreducibles in Γ. In each case, computer calculations show that the quotient of H 4 modulo the congruence generated by Γ is isomorphic to A 4 or B 4 . This shows that for each surjective homomorphisms from H 3 , there is a unique surjective homomorphisms from H 4 . Furthermore, Theorem 7.4 and Proposition 2.6 imply that there is no surjective lattice homomorphism from H 4 to B 4 whose associated congruence contracts qr and rqrq. Thus we have the following theorems: Theorem 7.5. There are exactly nine surjective lattice homomorphisms from H 4 to A 4 that fix S pointwise: For each choice of distinct γ 1 , γ 2 ∈ {qr, qrq, qrqr} and distinct γ 3 , γ 4 ∈ {rq, rqr, rqrq}, there exists a unique such homomorphism whose associated congruence contracts γ 1 , γ 2 , γ 3 and γ 4 . Theorem 7.6. There are exactly eight surjective lattice homomorphisms from H 4 to B 4 : There is no surjective lattice homomorphism η : H 4 → B 4 whose associated congruence contracts qr and rqrq. For every other choice of γ 1 ∈ {qr, qrq, qrqr} and γ 2 ∈ {rq, rqr, rqrq}, there exists a unique such homomorphism whose associated congruence contracts γ 1 and γ 2 .

Lattice homomorphisms between Cambrian lattices
In this section, we show how the results of this paper on lattice homomorphisms between weak orders imply similar results on lattice homomorphisms between Cambrian lattices.
A Coxeter element of a Coxeter group W is an element c that can be written in the form s 1 s 2 · · · s n , where s 1 , s 2 , . . . , s n are the elements of S. There may be several total orders s 1 , s 2 , . . . , s n on S whose product is c. These differ only by commutations of commuting elements of S. In particular, for every edge r-s in the Coxeter diagram for W (i.e. for every pair r, s with m(r, s) > 2), either r precedes s in every reduced word for c or s precedes r in every reduced word for c. We use the shorthand "r precedes s in c" or "s precedes r in c" for these possibilities.
The initial data defining a Cambrian lattice are a finite Coxeter group W and a Coxeter element c. The Cambrian congruence is the homogeneous degree-2 congruence Θ c on W generated by the join-irreducible elements alt k (s, r) for every pair r, s ∈ S such that r precedes s in c and every k from 2 to m(r, s) − 1. Here, as in Section 3, the notation alt k (s, r) stands for the word with k letters, starting with s and then alternating r, s, r, etc. Equivalently, Θ c is the finest congruence with s ≡ alt m(r,s)−1 (s, r) for every pair r, s ∈ S such that r precedes s in c. The quotient W/Θ c is the Cambrian lattice. The Cambrian lattice is isomorphic to the subposet of W induced by the bottom elements of Θ c -classes. To distinguish between the two objects, we write W/Θ c for the Cambrian lattice constructed as a quotient of W and write Camb(W, c) for the Cambrian lattice constructed as the subposet of c induced by bottom elements of Θ c -classes.
The key result of this section is the following theorem, which will allow us to completely classify surjective homomorphisms between Cambrian lattices, using the classification results on surjective homomorphisms between weak orders. Theorem 8.1. Let η : W → W ′ be a surjective lattice homomorphism whose associated congruence Ψ is generated by a set Γ of join-irreducible elements. Let c = s 1 s 2 · · · s n be a Coxeter element of W and let c ′ = η(s 1 )η(s 2 ) · · · η(s n ) ∈ W ′ . Then the restriction of η is a surjective lattice homomorphism from Camb(W, c) to Camb(W ′ , c ′ ). The associated congruence is generated by Γ ∩ Camb(W, c).
To understand Theorem 8.1 correctly, one should keep in mind that η is a lattice homomorphism, not a group homomorphism, so that c ′ need not be equal to η(c).
(For example, in the case n = 2 of Theorem 6.1, the lattice homomorphism η σ sends the Coxeter element s 0 s 1 to s 1 .) The element c ′ is a Coxeter element of W ′ in light of Proposition 2.1 and Theorem 1.5.
Before assembling the tools necessary to prove Theorem 8.1, we illustrate the theorem by extending Example 1.1. Figure 3 indicates a congruence Ψ on B 3 such that the quotient B 3 /Ψ is isomorphic to S 4 . As mentioned just after Theorem 6.7, the corresponding surjective homomorphism is η δ : B 3 → S 4 . The congruence Ψ is generated by Γ = {s 0 s 1 s 0 , s 1 s 0 s 1 }.

Example 8.2. Recall that
Let c be the Coxeter element s 0 s 1 s 2 of B 3 . The congruence Θ c on B 3 is generated by {s 1 s 0 , s 1 s 0 s 1 , s 2 s 1 }, as illustrated in Figure 13.a. Let c ′ be the Coxeter element s 1 s 2 s 3 of S 4 . The Cambrian congruence Θ c ′ on S 4 is generated by the set {s 2 s 1 , s 3 s 2 }, as illustrated in Figure 13.b. The set {s 2 s 1 , s 3 s 2 } is obtained by applying η δ to each element of {s 1 s 0 , s 1 s 0 s 1 , s 2 s 1 } that is not contracted by Ψ. Figure 14 shows the Cambrian congruence Θ c on B 3 and the Cambrian congruence Θ c ′ on S n . Here, S 4 is represented, as in Figure 3.b, as the subposet of B 3 induced by bottom elements of the congruence Ψ. Figure 15 shows the Cambrian lattice Camb(B 3 , c) and the Cambrian lattice Camb(S 4 , c ′ ). The shaded edges of Camb(B 3 , c) indicate the congruence on Camb(B 3 , c) whose quotient is Camb(S 4 , c ′ ). This congruence is generated by the join-irreducible element s 1 s 0 s 1 , because Γ ∩ Camb(W, c) = {s 0 s 1 s 0 }.
The proof of Theorem 8.1 depends on general lattice-theoretic results, but also on special properties of the subposet Camb(W, c) of W . To explain these properties, we recall from [25] the characterization of bottom elements of Θ c -classes as the c-sortable elements of W .
To define c-sortable elements, we fix a reduced word s 1 · · · s n for c and construct, for any element w ∈ W , a canonical reduced word for w. Write a half-infinite word c ∞ = s 1 · · · s n |s 1 · · · s n |s 1 · · · s n | . . ., where the symbols " | " are dividers that mark the locations where the sequence s 1 · · · s n begins again. Out of all subwords of c ∞ that are reduced words for w, the (s 1 · · · s n )-sorting word for w is the one that is lexicographically leftmost, as a sequence of positions in c ∞ . For convenience, when we write (s 1 · · · s n )-sorting words, we include the dividers that occur between letters in the lexicographically leftmost reduced subword. Thus for example, if W = S 4 , the (s 1 s 2 s 3 )-sorting word for the longest element is s 1 s 2 s 3 |s 1 s 2 |s 1 , and the (s 2 s 1 s 3 )-sorting word for the longest element is s 2 s 1 s 3 |s 2 s 1 s 3 . Each (s 1 · · · s n )sorting word defines a sequence of subsets of S, by taking the elements between successive dividers, and this sequence depends only on c, not on s 1 · · · s n . An element w is c-sortable if this sequence of subsets is weakly decreasing. In general, the set of bottom elements of a congruence is a join-sublattice but need not be a sublattice.
As a special case of a general lattice-theoretic result (found, for example, in [27, Proposition 9-5.11]), a c-sortable element v represents a join-irreducible element of the Cambrian lattice W/Θ c if and only if v is join-irreducible as an element of W . The following simple lemma will be used in the proof of Theorem 8.1.
Lemma 8.4. If j is a c-sortable join-irreducible element and j * is the unique element covered by j, then j * is c-sortable.
Proof. If c = s 1 · · · s n and a 1 a 2 · · · a k is the (s 1 · · · s n )-sorting word for j, then a 1 a 2 · · · a k−1 is the (s 1 · · · s n )-sorting word for a c-sortable element x covered by j. But j * is the unique element covered by j, so j * = x, which is c-sortable.
We need one of the standard Isomorphism Theorems for lattices. (See, for example, [27,].) This is easily proved directly, or follows as a special case of the same Isomorphism Theorem in universal algebra. The notation [x] Θ stands for the Θ-class of x. The following lemma will also be useful. Proof. Identifying join-irreducible elements of L with join-irreducible congruences as before, the sets I andĨ are the ideals that Γ andΓ generate in Irr(Con(L)). The join-irreducible elements in the quotient L/Ψ are exactly the elements [j] Ψ where j ∈ (Irr(L) \ I). Since the set of join-irreducible elements contracted by Ψ ∨Ψ is I ∪Ĩ, each [j] Ψ is contracted by the congruence (Ψ ∨Ψ)/Ψ if and only if j is inĨ. Furthermore, a join-irreducible element j ∈ (Irr(L) \ I) is inĨ if and only if it is below some element ofΓ \ I. We see that (Ψ ∨Ψ)/Ψ is the congruence on L/Ψ generated by the set {[j] Ψ : j ∈Γ \ I}.
We now prove our key theorem.
In particular, there is a surjective homomorphism from W/Θ c to W/(Ψ ∨ Θ c ) whose associated congruence is generated by {[j] Θc : j ∈ (Γ ∩ Camb(W, c))}. Since x → [x] Θc is an isomorphism from Camb(W, c) to W/Θ c , and since W/(Ψ ∨ Θ c ) is isomorphic to Camb(W ′ , c ′ ), we conclude that there is a surjective homomorphism from Camb(W, c) to Camb(W ′ , c ′ ) whose associated congruence is generated by Γ ∩ Camb(W, c). We will show that this homomorphism is the restriction of η.
If x ∈ Camb(W ′ , c ′ ), then x is the bottom element of [x] Θ c ′ . Thus η −1 (x) is a Ψ-class in W containing the bottom element y of a (Ψ ∨ Θ c )-class. In particular, y is the bottom element of [y] Θc , or in other words y ∈ Camb(W, c). Since η(y) = x, we have shown that Camb(W ′ , c ′ ) ⊆ η (Camb(W, c)).
Since Camb(W, c) is a sublattice of W and η is a homomorphism, η(Camb(W, c)) is a sublattice of W ′ , and the restriction of η to Camb(W, c) is a lattice homomorphism from Camb(W, c) to η (Camb(W, c)). Let j ∈ Γ ∩ Camb(W, c). Then because j ∈ Γ, η contracts j, or in other words η(j) = η(j * ), where j * is the unique element of W covered by j. By Lemma 8.4, the element j * is also in Camb(W, c), so j * is also the unique element of Camb(W, c) covered by j. Thus the restriction of η to Camb(W, c) also contracts j in Camb(W, c). Since there exists a surjective lattice homomorphism from Camb(W, c) to Camb(W ′ , c ′ ) whose associated congruence is generated by Γ ∩ Camb(W, c), the image of the restriction of η can be no larger than Camb(W ′ , c ′ ), and we conclude that η(Camb(W, c)) = Camb(W, c). We have shown that the restriction of η is a surjective homomorphism from Camb(W, c) to Camb(W ′ , c ′ ) whose associated congruence is generated by Γ ∩ Camb(W, c).
With Theorem 8.1 in hand, we prove several facts that together constitute a detailed classification of surjective homomorphisms between Cambrian lattices.
An oriented Coxeter diagram is a directed graph (with labels on some edges) defined by choosing an orientation of each edge in a Coxeter diagram. Orientations of the Coxeter diagram of a finite Coxeter group W are in bijection with Coxeter elements of W . There is disagreement in the literature about the convention for this bijection. The definition of Cambrian lattices given here agrees with the definition in [23] if we take the following convention, which also agrees with the convention of [24,25]: To obtain a Coxeter element from an oriented diagram, we require, for each directed edge s → t, that s precedes t in every expression for c. However, the opposite convention is also common.
An oriented diagram homomorphism starts with the oriented Coxeter diagram encoding a Coxeter system (W, S) and a choice of Coxeter element c, then deletes vertices, decreases labels on directed edges, and/or erases directed edges, and relabels the vertices to obtain the oriented Coxeter diagram of some Coxeter system (W ′ , S ′ ) and choice c ′ of Coxeter element. We prove the following result and several more detailed results. The proof of Theorem 8.7 begins with a factorization result analogous to Theorem 1.5. Given J ⊆ S and a Coxeter element c of W , the restriction of c to W J is the Coxeter elementc of W J obtained by deleting the letters in S \ J from a reduced word for c. (Typically, the restriction is not equal to c J .) Recall from Section 2 the definition of the parabolic homomorphism η J . Recall from Section 1 that a compressive homomorphism is a surjective homomorphism that restricts to a bijection between sets of atoms. For Cambrian lattices, this is a surjective lattice homomorphism Camb(W, c) → Camb(W ′ , c ′ ) that restricts to a bijection between S and S ′ . As was the case for the weak order, the task is to understand compressive homomorphisms between Cambrian lattices. The following proposition is proved below as a special case of part of Proposition 8.12. Together with Theorem 8.8, it implies that every surjective lattice homomorphism between Cambrian lattices determines an oriented diagram homomorphism. Proposition 8.9. Suppose η : Camb(W, c) → Camb(W ′ , c ′ ) is a compressive homomorphism. Then m ′ (η(r), η(s)) ≤ m(r, s) for each pair r, s ∈ S. Also, if s 1 · · · s n is a reduced word for c, then c ′ = η(s 1 ) · · · η(s n ).
As we did for the weak order, in studying compressive homomorphisms between Cambrian lattices, we may as well take S ′ = S and let η fix each element of S. We will prove the following characterization of compressive homomorphisms. (1) There exists a unique homomorphism η : Camb(W, c) → Camb(W ′ , c ′ ) that is compressive and fixes S pointwise and whose associated congruence Θ contracts all of elements ofΓ. (2) Θ is generated byΓ.
(3) Every compressive homomorphism from Camb(W, c) to Camb(W ′ , c ′ ) that fixes S pointwise arises in this manner, for some choice ofΓ.
Theorem 8.1 implies that, given any compressive homomorphismη : W → W ′ , there is a choice ofΓ in Theorem 8.10 such that every element ofΓ is contracted by the congruence associated toη. The homomorphism η thus arising from Theorem 8.10 is the restriction ofη. The following theorem is a form of converse to these statements. (1) There exists a compressive homomorphism from W to W ′ whose restriction to Camb(W, c) is η. (2) LetΓ be the set of join-irreducible elements that generates the congruence associated to η. Given a compressive homomorphismη : W → W ′ such that all elements ofΓ are contracted by the congruence associated toη, the restriction ofη to Camb(W, c) is η.
We begin our proof of these classification results by pointing out some basic facts on surjective homomorphisms between Cambrian lattices, in analogy with Proposition 2.1. Proposition 8.9 is a special case of (5) and (6) in the following proposition.
Proof. The proof of Proposition 2.1 can be repeated verbatim to prove all of the assertions except (6). The latter is equivalent to the statement that η(r) precedes η(s) if and only if r precedes s, whenever r, s ∈ S have η(r) = 1 ′ and η(s) = 1 ′ . This statement follows immediately from (4) with J = {r, s}.
Taking η = η J and Γ = S \ J in Theorem 8.1, we have the following analog of Theorem 2.5 for Cambrian lattices: Theorem 8.13. Let c be a Coxeter element of W, let J ⊆ S and letc be the Coxeter element of W J obtained by restriction. Then η J restricts to a surjective lattice homomorphism from Camb(W, c) to Camb(W J ,c). The associated lattice congruence on Camb(W, c) is generated by the set S \ J of join-irreducible elements.
To prove Theorem 8.8, the proof of Theorem 1.5 can be repeated verbatim, with Proposition 8.12 replacing Proposition 2.1 and Theorem 8.13 replacing Theorem 2.5. We prove Theorems 8.10 and 8.11 together.
Proof of Theorems 8. 10 and 8.11. We first claim that there is a compressive homomorphismη : W → W ′ fixing S and a generating set Γ for the associated congruence such thatΓ = Γ ∩ Camb(W, c). Arguing as in previous sections, Theorems 3.1 and 4.2 reduce the proof of the claim to the case where W and W ′ are irreducible and their diagrams coincide except for edge labels. Looking through the type-by-type results of Sections 5-7, we see that in almost every case, the claim is true for a simple reason: For each choice ofΓ, there is a surjective homomorphism from W to W ′ whose associated congruence is homogeneous of degree 2, with a generating set that includesΓ. The only exceptions come when (W, W ′ ) is (H 3 , B 3 ) or (H 4 , B 4 ) and one of the following cases applies: Case 1: q precedes r andΓ = {qr}. In this case, in light of Case 2 of the proof of Theorem 7.4, we need to verify that qrqsrqrs ∈ H 3 is not qrs-sortable and not qsr-sortable. But qrqsrqrs has only one other reduced word, qrsqrqrs, so its qrssorting word is qrs|qr|qrs, and thus it is not qrs-sortable. Similarly, the qsr-sorting word for qrqsrqrs is qr|qsr|qr|s, so it is not qsr-sortable. The claim is proved in auxiliary Lie algebrag(A) using generators and relations. The Lie algebrag(A) decomposes as a direct sum n − ⊕ h ⊕ n + of complex vector spaces. We won't need details on h here, but its dual contains linearly independent vectors α 1 , . . . , α n called the simple roots. The summand n + decomposes further as a (vector space) direct sum with infinitely many summandsg α , indexed by nonzero vectors α in the nonnegative integer span Q + of the simple roots. Similarly, n − decomposes into summands indexed by nonzero vectors in the nonpositive integer span of the simple roots. Furthermore, n + is freely generated by elements e 1 , . . . , e n and thus is independent of the choice of A (as long as A is n × n). Similarly, n − is freely generated by elements f 1 , . . . , f n and is independent of the choice of A.
There is a unique largest ideal r ofg(A) whose intersection with h is trivial, and this is a direct sum r − ⊕ r + of ideals with r ± = r ∩ n ± . The Kac-Moody Lie algebra g(A) is defined to byg(A)/r. The Lie algebra g(A) inherits a direct sum decomposition n − ⊕ h ⊕ n + , and the summands n ± decompose further as n − = 0 =α∈Q+ g −α and n + = 0 =α∈Q+ g α .
A root is a nonzero vector α in Q + ∪ (−Q + ) such that g α = 0. The (Kac-Moody) root system associated to A is the set of all roots. While n − and n + are independent of the choice of A, the ideal r depends on A (in a way that is not apparent here because we have not given the presentation ofg(A)). Thus, writing r(A) for the ideal associated to A, Proposition 1.10 follows immediately from this claim: If A and A ′ are Cartan matrices such that A dominates A ′ , then r(A) ⊆ r(A ′ ).
This claim follows immediately from a description of the ideal r in terms of the Serre relations. Recall that for x in a Lie algebra g, the linear map ad x : g → g is y → [x, y]. In [13,Theorem 9.11], it is proved that the ideal r + is generated by {(ad ei ) 1−aij (e j ) : i = j} and that r + is generated by {(ad fi ) 1−aij (f j ) : i = j}.
Although we have proved Proposition 1.10 in full generality, we pause to record the following proof due to Hugh Thomas (personal communication, 2018) in the case where A is symmetric.
Quiver-theoretic proof of Proposition 1.10 for A symmetric. Suppose Q is a quiver Q without loops (i.e. 1-cycles) with n vertices, suppose v ∈ Z n , and fix an algebraically closed field. We associate a symmetric Cartan matrix A to Q by ignoring the direction of edges and taking a ij to be the total number of edges connecting vertex i to vertex j.
Kac [12,Theorem 3] (generalizing Gabriel's Theorem) proved that v is the dimension vector of an indecomposable representation of Q if and only if v is the simple-root coordinate of a positive root in the root system Φ(A).
Given symmetric Cartan matrices A and A ′ with A dominating A ′ , we can construct a quiver Q ′ associated to A ′ and add arrows to it to obtain a quiver Q associated to A. If v is the simple root coordinates of a root in Φ(A ′ ), then if we start with an indecomposable representation of Q ′ with dimension vector v and assign arbitrary maps to the arrows in Q that are not in Q ′ , the result is still indecomposable, so v is the simple root coordinates of a root in Φ(A). The dual root system Φ ∨ (A) consists of all co-roots associated to the roots in Φ(A). This is a root system in its own right, associated to A T . Since A dominates A ′ if and only if A T dominates (A ′ ) T , the following proposition is immediate from Proposition 1.10.
Proposition 9.1. Suppose A and A ′ are symmetrizable Cartan matrices such that A dominates A ′ . If Φ ∨ (A) and Φ ∨ (A ′ ) are both defined with respect to the same simple co-roots α ∨ i , then Φ ∨ (A) ⊇ Φ ∨ (A ′ ) and Φ ∨ + (A) ⊇ Φ ∨ + (A ′ ). Suppose A dominates A ′ and suppose (W, S) and (W ′ , S) are the associated Coxeter systems. Let A and A ′ be the associated Coxeter arrangements, realized so that the simple co-roots are the same for the two arrangements. (This requires that two different Euclidean inner products be imposed on R n .) Proposition 9.1 implies that A ′ ⊆ A, so that in particular, each region of A is contained in some region of A ′ . Since the regions of A are in bijection with the elements of W and the regions of A ′ are in bijection with the elements of W ′ , this containment relation defines a surjective map η : W → W ′ .
Theorem 9.2. The map η, defined above, is a surjective lattice homomorphism from the weak order on W to the weak order on W ′ .
Proof. As in the proof of Proposition 1.10, we may restrict our attention to the case where A is irreducible and barely dominates A ′ . We first consider the case where A ′ is obtained by erasing a single edge e in the Dynkin diagram of A. Let E = {e}. Recall from Section 3 that the map η E maps w ∈ W to (w I , w J ) ∈ W I × W J = W ′ , where I and J are the vertex sets of the two components of the diagram of A ′ . The set of reflections in W ′ equals the set of reflections in W I union the set of reflections in W J . Since inv(w I ) = inv(w) ∩ W I and inv(w J ) = inv(w) ∩ W J , two elements of W map to the same element of W ′ if and only if the symmetric difference of their inversion sets does not intersect W I ∪ W J . Comparing with the edge-erasing case in the proof of Proposition 1.10, and noting that both maps fix W ′ , we see that η = η E . Theorem 3.2 now says that η is a surjective homomorphism.
The case where A is of type G 2 is easy and we omit the details. Now suppose A is of type C n , so that the dual root system Φ ∨ (A) is of type B n . Using a standard realization of the type-B root system, reflections in W correspond to positive co-roots as follows: A co-root e i corresponds to (−i i), a co-root e j − e i corresponds to (−j − i)(i j), and a co-root e j + e i corresponds to (i − j)(j − i). The positive co-roots for A ′ are the roots α 1 + α 2 + · · · + α j = e j for 1 ≤ j ≤ n and α i + α i+1 + · · · + α j = e j − e i−1 for 2 ≤ i ≤ j ≤ n. Two adjacent A-regions are contained in the same A ′ -region if and only if the hyperplane separating them is in A \ A ′ . Thus, in light of Proposition 6.2, we see that η has the same fibers as the map η σ of Section 6.1, which is a surjective lattice homomorphism by Theorem 6.1. Writing, as before, τ for the inverse map τ to η σ , the inversions of a permutation π correspond to the inversions t of τ (π) such that H t ∈ A ′ . It follows that η = η σ .
When A is of type B n , so that Φ ∨ (A) is of type C n , the argument is the same, using Proposition 6.5 instead of Proposition 6.2.
The positive co-roots for the A ′ , as a subset of these co-roots, are {e 1 , e 2 , e 3 } ∪ {e 2 − e 1 , e 3 − e 2 , e 3 − e 1 } ∪ Computer calculations show that the surjective homomorphism η found in Case 4 of the proof of Theorem 7.1 has the property that two elements of W , related by a cover in the weak order, map to the same element of W ′ if and only if the reflection that relates them corresponds to a co-root not contained in the subset Φ ∨ (A ′ ). The two maps coincide by the same argument given in previous cases.