Cataland: Why the Fuss?

The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuss-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuss-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras.


Introduction
Fix a finite Coxeter system (W, S) and a (standard) Coxeter element c ∈ Wthat is, a product of the simple generators S in any order. There are three noncrossing Catalan objects associated to this data [Rea07a,Rea07b,BRT12,PS11]: • the c-noncrossing partitions NC(W, c); • the c-sortable elements Sort(W, c); and • the c-cluster complexes Asso(W, c). We use the term "noncrossing" here for historical reasons-noncrossing partitions generalize noncrossing set partitions to all finite Coxeter systems and all standard Coxeter elements. We also attach this term to the other families, since they are in uniform bijection with the noncrossing partitions, and to distinguish them from the nonnesting Catalan objects, which we do not consider in this paper. We denote the cluster complex by Asso(W, c) in order to emphasize that it generalizes the well-studied dual associahedron.
Although no uniform proof is currently known, these noncrossing families are counted by the Catalan number of type W where d 1 ≤ d 2 ≤ . . . ≤ d n are the degrees and h := d n = ord(c) is the Coxeter number of W . These noncrossing Catalan objects have two natural lattice structures, each of which has an associated cyclic group action: • The c-noncrossing partition lattice NCL(W, c) and its anti-automorphism of order 2h given by the Kreweras complement Krew c ; and • the c-Cambrian lattice Camb(W, c) and its underlying graph automorphism of order h + 2 given by the Cambrian rotation Camb c . The noncrossing partitions, their lattices and Kreweras complements, and the cluster complex and Cambrian rotation have been generalized in the literature by introducing a nonnegative integral parameter m. Since many generalizations exist, and since many names in the theory already carry the term "generalized", we refer to this particular generalization as an m-eralization. The objects associated with these m-eralizations are counted by the Fuß-Catalan number of type W More precisely, • S. Fomin and N. Reading defined and studied the m-eralized c-cluster complex Asso (m) (W, c) for a bipartite Coxeter element c in [FR05]; and • D. Armstrong defined and studied the m-eralized c-noncrossing partitions NC (m) (W, c) for any Coxeter element c in [Arm06], generalizing P. Edelman's construction from [Ede80] to all finite Coxeter groups. There are several components missing from this story. The aim of this paper is to complete the m-eralization of noncrossing Catalan objects using the spherical Artin group corresponding to the finite Coxeter system (W, S).
Coxeter-sortable elements. The most glaring omission is that no m-eralization of N. Reading's c-sortable elements [Rea07b] has appeared in the literature. It is straightforward to extract a definition involving chains of sortable elements by combining D. Armstrong's definition of m-eralized c-noncrossing partitions and N. Reading's shard order (see Section 4.7), but such a definition is unsatisfactory without a corresponding m-eralization of the weak order. At the heart of this paper is the observation that such an m-eralization is given by the interval [e, w m • ] in the weak order of the positive Artin monoid B + = B + (W ). These intervals have been previously studied by P. Dehornoy in an enumerative context [Deh07], and we study some of their properties in Section 2.8.
Definition 1.1. W (m) := w ∈ B + (W ) : w ≤ w m • ∈ B + (W ) . There is a natural bijection between the elements of W and W (1) . Furthermore, W (m) is a rank-symmetric lattice under the weak order, and we recover the usual weak order when m = 1. In Section 4.3, we provide the missing m-eralization of the c-sortable elements as a lift of N. Reading's definition from W to W (m) . Definition 1.2. Let c ∈ W be a Coxeter element. An element w ∈ W (m) is csortable if the c-sorting word w(c) for w defines a decreasing sequence of subsets of positions in c. Denote the set of all such m-eralized c-sortable elements by Sort (m) (W, c).
We refer to Section 2.1 for the definition of the c-sorting word w(c). We provide three characterizations of Sort (m) (W, c) in Definition 4.5, Proposition 4.9, and in Definition 4.13, extending N. Reading's characterization of c-sortable elements when m = 1, as given in [Rea07a,Sections 2 and 4]. The relationship between chains of c-sortable elements in shard order and Sort (m) (W, c) turns out to be analogous to the relationship between chains of noncrossing partitions and delta sequences, as explained in Theorem 4.56.
Cluster complexes. The second missing component is that there is no simple combinatorial definition of the m-eralized c-cluster complex when the Coxeter element c is not bipartite. We supply this definition in Section 5.4, giving an meralized c-compatibility relation. We prove the existence and uniqueness of this relation using an m-eralization of the subword complex approach to c-cluster complexes given in [CLS14,PS11]. Using this definition, we recover in Proposition 5.25 that Asso (m) (W, c) is vertexdecomposable, and hence shellable. With the m-eralizations of noncrossing partitions, sortable elements, and subword complexes in hand, we prove the following theorem and several refinements in Theorems 4.34, 4.56, 5.33 and 5.37. By "explicit," we mean that we provide a bijection, rather than just proving existence of a bijection by a counting argument. The term "uniform" specifies that the description of the bijection does not use the classification theorem of finite Coxeter systems.
Cambrian lattices. Although the exchange graph of S. Fomin and N. Reading's m-eralized c-cluster complex can be used to define a Cambrian graph for bipartite Coxeter elements, no corresponding poset has been considered in the literature for m > 1. In particular, no orientation of the exchange graph was known to be a lattice. In Sections 3.4, 4.6 and 5.7, we close this third gap with definitions of the m-eralized c-Cambrian lattice on each of NC (m) (W, c), Sort (m) (W, c), and Asso (m) (W, c). Our construction m-eralizes N. Reading's Cambrian lattices, which are themselves generalizations of the classical Tamari lattices.
For NC (m) (W, c) and Asso (m) (W, c), we construct these posets as the transitive closures of the objects under certain flips. Exactly as in the case m = 1, the structure on Sort (m) (W, c) is inherited from the weak order on W (m) . This yields the following theorem which is proven in Theorems 4.40, 4.42 and 5.42.
Theorem 1.5. The restriction of the weak order to Sort (m) (W, c) is a lattice. It is isomorphic to the increasing flip posets of NC (m) (W, c) and of Asso (m) (W, c).
We call this common lattice the m-eralized c-Cambrian lattice. The intuition behind this construction on NC (m) (W, c)-which appears to be new even when m = 1-is given in Section 3.4, where it naturally arises from transporting a construction of weak order via descent and ascent sets to the noncrossing partitions. Another intuition comes from the isomorphism between shard order restricted to c-sortable elements and the noncrossing partition lattice: just as D. Armstrong considered component-wise absolute order on chains in NCL(W, c) [Arm06], our meralized c-Cambrian lattices can be seen to be component-wise weak order on the corresponding chains in Shard(W ).  In summary, we place the program of m-eralizing noncrossing Coxeter-Catalan combinatorics in the context of the corresponding positive Artin monoid. For the reader's convenience, Figure 1 displays the Fuß-Catalan objects we consider, and the bijections between the various objects.
At least in the classical types, our definitions naturally extend to Coxetertheoretic constructions of noncrossing objects in the framework of D. Armstrong's program of "rational Catalan combinatorics" [ARW13,ALW14], which we plan to explore in the future. In a subsequent paper, we will study local versions of the Kreweras complement and of Cambrian rotation, and we will relate these local actions to the study of nonnesting partitions.

m-eralized weak order
In this section, we first recall some background on finite Coxeter groups and their Artin groups. Further background can be found in [BS72, DDG + 13, Hum90]. We will then consider inversion sequences, introduce subword complexes and dual subword complexes, and recall N. Reading's definition of Coxeter-sorting words. Finally, we define an m-eralized version of the weak order and study some of its properties.
2.1. Coxeter and Artin groups. A finite Coxeter system (W, S) of rank n = |S| is a finite group W together with a distinguished subset S ⊆ W of generators and a presentation W = S : s 2 = e, (st) m(s,t)/2 = (ts) m(t,s)/2 for s, t ∈ S with s = t , for integers m(s, t) = m(t, s) ≥ 2, and where we set The elements in the set S are called simple generators or simple reflections. The relations sts · · · = tst · · · are called braid relations, and replacing sts · · · by tst · · · in a word in S is called a braid move. Observe that for s, t ∈ S, we have m(s, t) = 2 if and only if s and t commute in W . It follows from the classification of finite Coxeter systems that we have the decomposition S = S L ⊔S R , where all reflections in S L pairwise commute, as do those in S R . To not have to distinguish various cases later, we fix such a decomposition throughout the paper.
The set of reflections in W is defined to be R := s w : s ∈ S, w ∈ W , where we write u w := wuw −1 . The spherical Artin system (B(W ), S) corresponding to the finite Coxeter system (W, S) is the group B(W ) given by (a formal copy of) the generators S and the presentation B(W ) = S : (ss ′ ) m(s,s ′ )/2 = (s ′ s) m(s ′ ,s)/2 for s, s ′ ∈ S with s = s ′ .
When W is a symmetric group, B(W ) is called a braid group. For ease of notation, we will not explicitly distinguish between the simple generators of W and of B(W ). Rather than studying B(W ), we will restrict ourselves to the positive Artin monoid B + (W ) (or simply B + if there is no confusion), which is the submonoid of B(W ) generated by S.
Example 2.1. To illustrate our definitions and results, we will use the running example of the symmetric group on three letters. This group is the Coxeter group A 2 with simple generators S = {s, t} that correspond to the simple transpositions s = (12) and t = (23). The third reflection is denoted by u = (13), giving R = {s, u, t} = (12), (13), (23) .
Beside these reflections and the identity element e, the group A 2 also contains the two "rotations" (123) = st = tu = us and (321) = ts = ut = su.
The positive Artin monoid B + (A 2 ) consists of all words in s and t, subject to the relation sts = tst.

2.2.
Coxeter length, support, and absolute length. The (Coxeter) length of an element w in the group W or in the positive Artin monoid B + is the length ℓ(w) of a shortest expression for w as a product of the generators in S. An S-word s 1 · · · s p with s 1 , · · · , s p ∈ S is reduced if w = s 1 · · · s p and p = ℓ(w). Figures 2  and 3 illustrate several examples. We use Roman letters to indicate that we consider words in S rather than elements in W or in B + .
It is classical that any reduced word of an element in W may be transformed to any other by a sequence of braid moves [BB05, Theorem 3.3.1]. A reduced word for an element w ∈ W gives rise to a corresponding element in B(W ); since B(W ) contains the braid relations, any two reduced words for w ∈ W specify the same element in B(W ). The unique longest element in W is denoted by w • , and we will also use the notation w • to refer to the corresponding element in B + (W ). This element is called the fundamental element in Garside theory. We denote its length by N = ℓ(w • ) = |R|.
We define the support supp(w) for w ∈ W to be the set {s 1 , . . . , s p } ⊆ S of simple reflections contained in any reduced word s 1 · · · s p for w. Analogously, we define the support supp(w) for w ∈ B + (W ), since all words for w ∈ B + (W ) are-by definition-related by a sequence of braid moves. We again refer to Figures 2 and 3 for examples.
Instead of considering words in simple reflections for elements in W , it is natural to also consider words in general reflections. The absolute length of an element w in W is the length ℓ R (w) of a shortest expression for w as a product of reflections. An R-word r 1 · · · r p with r 1 , . . . , r p ∈ R is reduced if w = r 1 · · · r p and p = ℓ R (w). We now only refer to Figure 2 for examples. See [BKL98] for a construction of the braid group and [Bes03] for constructions of spherical Artin groups using a related construction.
2.3. Finite reflection groups. We briefly review the geometry of Coxeter systems. Let V be an n-dimensional Euclidean vector space. For v ∈ V \ {0}, we denote by s v the reflection interchanging v and −v while fixing the orthogonal hyperplane pointwise. We consider a finite (real) reflection group W acting on V , i.e., a finite group generated by the reflections R it contains. We assume without loss of generality that the intersection of all the corresponding reflecting hyperplanes is {0}. Fix a connected component-the fundamental chamber-of the complement of the reflecting hyperplanes. The simple reflections of W with respect to this choice of a fundamental chamber then are the n reflections orthogonal to its facet-defining hyperplanes. The set S ⊆ R of simple reflections generates W , and the pair (W, S) then forms a Coxeter system. Moreover, every Coxeter system may be obtained in this way.
2.4. Root systems. Let W be a finite reflection group with a fixed choice of a fundamental chamber. For simple reflections s, t ∈ S, denote by m st the order of the product st ∈ W . Fix a generalized Cartan matrix for (W, S), i.e., a matrix (a st ) s,t∈S such that a ss = 2, a st ≤ 0, a st a ts = 4 cos 2 ( π mst ) and a st = 0 ⇔ a ts = 0 for all s = t ∈ S. We can associate to each simple reflection s a simple root α s ∈ V , orthogonal to the reflecting hyperplane of s and pointing toward the half-space containing the fundamental chamber, so that s(α t ) = α t −a st α s for all s, t ∈ S. The set of all simple roots is denoted by ∆ := {α s : s ∈ S}. The orbit Φ := {w(α s ) : w ∈ W, s ∈ S} of ∆ under W is a root system for W . It is invariant under the action of W and contains precisely two opposite roots orthogonal to each reflecting hyperplane of W . When discussing finite Coxeter systems, we always think of the Coxeter system together with a fixed root system associated to it, since there is not necessarily a unique root system associated to a finite reflection group W .
The set ∆ of simple roots forms a linear basis of V (since we assumed W to act essentially on V ). The root system Φ is the disjoint union of the positive roots Φ + := Φ ∩ R ≥0 [∆] (non-negative linear combinations of simple roots) and the negative roots Φ − := −Φ + . We write β > 0 or β < 0 to indicate if β is a positive or a negative root, and we denote by |β| the positive root in {±β}. Each reflecting hyperplane is orthogonal to one positive and one negative root. For a reflection r ∈ R, we set α r to be the unique positive root orthogonal to the reflecting hyperplane of r, so that r = s αr .
Example 2.2. The Coxeter group A 2 considered above can be realized as the dihedral group of isometries of a regular triangle. Its root system consists of the simple and positive roots ∆ = {α, β} and Φ + = {α, γ, β} with α = e 2 − e 1 , β = e 3 − e 2 , and γ = e 3 − e 1 = α + β and such that The Coxeter group W is said to be crystallographic if it stabilizes a lattice in V when considered as a finite reflection group. This can only happen if all entries of the Cartan matrix are integers, or, equivalently, if m(s, t) ∈ {2, 3, 4, 6} for all s, t ∈ S with s = t. Reciprocally, if all entries of the Cartan matrix are integers, then the lattice generated by the simple roots ∆ is fixed by the Coxeter group W .
A particular choice of a reduced expression s 1 · · · s ℓ for w ∈ W induces a total order inv(s 1 · · · s ℓ ) on the positive roots inv(w), and a reduced expression of the longest element w • induces an order inv(w • ) on Φ + = inv(w • ). Such orders on Φ + are called root orders. For simplicity of notation, we also refer with this term to the analogous orders on R induced by inv R (w • ). We refer to [Dye93] for further details on root orders.
Given any S-word Q = s 1 · · · s p , one can still consider the corresponding inversion sequence α s1 , s 1 (α s2 ), . . . , s 1 · · · s p−1 (α sp ) . Since such sequences contain negative roots, and possibly even repeated roots, if the word Q is not reduced, we now consider colored inversion sequences on colored positive roots, which are simply positive roots together with a color given by a nonnegative integer. A simple reflection s ∈ S acts on a colored positive root as s(β (k) ) = [s(β)] (k) if β = α s and s(β (k) ) = β (k+1) if β = α s . An element w ∈ W therefore acts on colored positive roots as The colored inversion sequence of Q is given by where β (mi) i is given by s 1 · · · s i−1 (α (0) si ). In other words, β i = |s 1 · · · s i−1 (α si )| ∈ Φ + and m i is given by the number of times s j · · · s i−1 (α si ) ∈ {±α sj } for j = 1, 2, . . . , i− 1, or, equivalently, the number of times s j · · · s i−1 (α si ) and s j+1 · · · s i−1 (α si ) have different signs. For ease of notation and agreement with the above definition for reduced S-words, we may suppress the superscripts if s 1 · · · s p is reduced.
Example 2.3. The inversion sequence in A 2 for the reduced S-word sts of the element u = sts is inv(sts) = α, γ, β, and the colored inversion sequence of its with reflection sequence inv R (stssts) = (s, u, t, t, u, s).
2.6. Subword complexes and dual subword complexes. We next consider certain natural simplicial complexes associated to words in S or in R and elements in W .
Definition 2.4. Let Q be a word in S, let w ∈ W , and let a = ℓ(w) + 2g for some g ≥ 0. The subword complex Sub S (Q, w, a) is the simplicial complex with facets being all subsets of (positions of) letters in Q whose complements yield an S-word for w of length a.
Definition 2.5. Let Q be a word in R, let w ∈ W , and let a = ℓ R (w) + 2g for some g ≥ 0. The dual subword complex Sub R (Q, w, a) is the simplicial complex with facets being all subsets of (positions of) letters in Q that yield an R-word for w of length a.
The definition of subword complexes for g = 0 was given by A. Knutson and E. Miller in [KM04], where they provided various properties of such subword complexes. In this case, we drop the parameter a from the notation, and we write Sub S (Q, w) and Sub R (Q, w), respectively. For w = w • , such subword complexes were studied in [CLS14,PS11]. After discussing the duality between the two notions of subword complexes, we will extend some properties from these two references to the case of general parameter g ≥ 0.
Subword and dual subword complexes are indeed dual to each other in the following sense.
Proposition 2.6. Let Q = s 1 · · · s p be a word in S and let w ∈ W . Let a = ℓ(w)+2g for some g ≥ 0, b = p − a, and w ′ = (ws p · · · s 1 ) −1 . Then the two complexes Sub S (Q, w, a) and Sub R (inv R (Q), w ′ , b) coincide.
Proof. The proof follows the same lines as the proof of [IS10, Lemma 3.2] and of [CLS14, Proposition 2.8]. Let inv R (Q) = r 1 · · · r p and let 1 ≤ i 1 ≤ . . . ≤ i b ≤ p be an increasing sequence of positions. This is a facet of Sub S (Q, w, a) if and only if s 1 · · ·ŝ i1 · · ·ŝ i b · · · s p = w.
Since the left-hand side equals r i b · · · r i1 s 1 · · · s p , the set {i 1 , . . . , i b } is a facet of Sub S (Q, w, a) if and only if which is equivalent to saying that {i 1 , . . . , i b } is a facet of Sub R (inv R (Q), w ′ , b).
Example 2.7. Let Q = stststst be the word in simple reflections in A 2 , let w = e = w 2 • , and let a = 6. The reflection sequence corresponding to Q is then inv R (Q) = (s, u, t, s, u, t, s, u) .
The 12 facets of the subword complex Sub S (Q, w, a) can be found in Figure 12 on page 46.
For Q = s 1 · · · s p , w ∈ W , and a = ℓ(w) + 2g, the facets of the subword complex Sub S (Q, w, a) are specified by those sequences i 1 < . . . < i p−a such that We now consider a slight variation of the colored inversion sequence, where the simple reflections s i1 , . . . , s ip−a are omitted when computing β of all colored positive roots r I (i) = β (mi) i associated to the elements of I in the above procedure. We call the colored positive root r I (i) the root vector of I at position i. Observe that the root configuration has a natural order induced by the order of the (positions of) letters in I. We thus consider the root configuration throughout the paper as an ordered set, and whenever we write a union of colored roots in a root configuration, we mean the disjoint union ordered by concatenation of the individual orderings.
Example 2.8. The root configurations of the facets of the subword complex in Example 2.7 can be found in Figure 12 on page 46.
The role of the root configuration for subword complexes is the same in the present generality as it is for g = 0. The original proofs of the following lemmas do not rely on the assumption that the word for w inside Q is reduced, and therefore generalize verbatim.
Lemma 2.9 ([PS11, Lemma 3.3(2)]). Let I be any facet of the subword complex Sub S (Q, w, a), and let r I (i) = β (k) . If J is a facet of Sub S (Q, w, a) adjacent to I (i.e., I \ i = J \ j), the position j is one of the positions in the complement of I for which r I (j) = β (ℓ) for k = ℓ. Moreover, i < j if and only if k < ℓ.
In the situation of this lemma, we say that the facets I and J are connected by a flip. Such a flip from I to J is increasing if i < j and decreasing otherwise. Finally, the direction of this flip from I to J is given by r I (i). Lemma 2.9 states that an increasing flip is also characterized by I \ i = J \ j with r I (i) = β (k) and r I (j) = β (ℓ) with k < ℓ. We conclude that the Hasse diagram of the increasing flip poset Γ(Q, w, m) is given by those edges I \ i = J \ j in the increasing flip graph G(Q, w, m) for which r I (i) = β (k) and r I (j) = β (k+1) for some k.
Example 2.12. Figure 13 on page 53 shows the increasing flip poset Γ(Q, w, m) for the S-word Q = stststst in A 2 , the element w = e = w 2 • , and a = 6 with increasing flips labeled by their direction.
Remark 2.13. There is an alternative generalization of subword complexes which computations suggest also works for our applications, although we do not pursue its development here. Let Q = s 1 · · · s p and let w ∈ B + (W ). The Artin subword complex Sub B S (Q, w) is the simplicial complex with facets being all subsets of (positions of) letters in Q that yield an S-word for w. Then there is an injection from the Artin subword complex Sub B S (Q, w) to the subword complex Sub S (Q, w, ℓ(w)), where w is the projection of w into W . We have chosen the above approach for the duality in Proposition 2.6.
It was shown in [CLS14, Section 3.3] that certain subword complexes with g = 0 are canonically isomorphic. The analogous statements hold in this more general context with the same proofs. We call two S-words Q and Q ′ commutation equivalent or equal up to commutation if they are obtained from each other by a sequence of interchanging consecutive commuting simple reflections. Two consecutive letters commute if and only if the corresponding reflections in inv R (Q) and inv R (Q ′ ) commute. Accordingly, we call two words Q and Q ′ in R commutation equivalent or equal up to commutation if they are obtained from each other by a sequence of interchanging consecutive commuting reflections.
Proposition 2.14. Let Q, Q ′ be commutation equivalent S-words (resp. R-words). Then the two complexes Sub S (Q, w, a) and Sub S (Q ′ , w, a) (resp. Sub R (Q, w, b) and Sub R (Q ′ , w, b)) are canonically isomorphic.
In the case of w = w m • ∈ {e, w • } ⊆ W for some integer m, there is the following further rotational symmetry. Observe that s (w m • ) = w m • sw m • ∈ W is again a simple reflection for any s ∈ S. The word Q ′ in the following proposition is therefore again a word in S.
Proposition 2.15. Let Q = s 1 · · · s p be an S-word and let Q ′ = s 2 · · · s p s Then Sub S (Q, w m • , a) and Sub S (Q ′ , w m • , a) are canonically isomorphic. Proofs of Propositions 2.14 and 2.15. The proofs given in [CLS14, Propositions 3.8 and 3.9] extend directly to these more general cases.
This more general construction of subword complexes does not guarantee vertexdecomposability for g > 0, as seen in the following example.
Example 2.16. Let Q = s 1 s 1 s 2 s 2 in type A 1 ×A 1 . Then the facets of Sub S (Q, e, 2) are given by {12, 34}. Since this complex is not connected, it is neither shellable nor vertex-decomposable.
We remark that the Artin subword complexes of Remark 2.13 are also not vertexdecomposable in general, although Proposition 2.15 still holds since for w m • ∈ B + and s ∈ S, we still have s (w m • ) ∈ S. Example 2.17. Let Q = tsstst, and let w = ssts = stst = tstt ∈ B + (W ). Then, as an Artin subword complex, is not an Artin subword complex. Using a slightly more complicated example, one can also show that vertex-decomposability does not even hold in the special case w = w m • . 2.7. Coxeter elements and sorting words. A (standard) Coxeter element c for (W, S) and for (B(W ), S) is defined to be the product of the simple reflections S in any order. Given the fixed bipartition S = S L ⊔ S R discussed in Section 2.1, a bipartite Coxeter element is the product of the reflections in S L followed by the product of the reflections in S R . As the Dynkin diagram for a finite Coxeter group W is a tree, we have that all Coxeter elements in W are conjugate. We denote their common order by the Coxeter number h. The rank n of W , the number N = ℓ(w • ) = |Φ + |, and the Coxeter number are related by N = nh/2.
Let c = s 1 · · · s n be a particular reduced word for a Coxeter element c, and let w be an element in W (or in B + ). The c-sorting word w(c) for w is then defined to be the lexicographically first subword of c ∞ = (s 1 · · · s n ) ∞ which is a reduced expression for w. To emphasize the different copies of s 1 · · · s n , we may separate them by vertical bars, as in Figure 2 on page 12. Observe that although the c-sorting word is attached to a particular reduced word for c-rather than to c itself-the property that all reduced words for c are related by commutation relations implies that the corresponding sorting words are also equal up to commutations. Finally, we write ≤ c for the root order inv(w • (c)) on Φ + and analogously for the root order inv R (w • (c)) on R.
The following lemma collects a few properties of sorting words and their reflection sequences. . Let c = s 1 · · · s n be a Coxeter element for a finite Coxeter system (W, S) with s = s 1 initial in c, let c ′ = s 2 · · · s n s 1 , and let c ′′ = s w• n · · · s w• 1 . For a word Q in S let Q denote the word obtained from Q by replacing every letter s in Q by s w• (which is again in S). Then (1) the sequence w • (c) starts with a copy of c; (2) the sequence w • (c) ends, up to commutations, with a copy of c; (3) the sequences w • (c) and w • ( c) coincide up to commutations; (4) the sequences w • (c) w • (c) and c h coincide up to commutations; (5) the sequence w • (c ′′ ) is the reverse of w • (c) up to commutations; (6) the sequence w • (c ′ ) is obtained from the sequence w • (c) by removing the initial s 1 , and appending a final s w• 1 , again up to commutations; and (7) if t 1 , t 2 ∈ R ∩ W s then α t1 < c α t2 ⇔ α t1 < s −1 cs α t2 ⇔ α t1 < s −1 c α t2 in W s , where W s is the maximal parabolic subgroup generated by S \ {s}.
Following Item 4, we define, for later convenience, where the roots in the ith copy of inv(w • (c)) have color i − 1. We note that the later Proposition 4.17 will generalize Item 4 to general sortable elements of B + .
2.8. m-eralized weak order and Garside factorizations. The (right) weak order on W (resp. B + ) is the partial order Weak(W ) (resp. Weak( If w has such a factorization uv = w, we say that u left-divides w and v rightdivides w. We may write wv −1 for u and u −1 w for v, even in B + . A letter s ∈ S is called initial (resp. final) in an element w in the group W or in the Artin monoid B + , if s left-divides (resp. right-divides) w.
It is well-known that the right weak order in W can also be described in terms of inversion sets as The left descent set, right descent set, left ascent set, and right ascent set of w ∈ W or of w ∈ B + by  We refer to Figure 2 for several examples.
Proof. Recall that if s, t ∈ des L (u) for u ∈ W or for u ∈ B + (W ), then s ∨ t = sts · · · ℓ(s∨t) factors = tst · · · ℓ(s∨t) factors is a left factor of u [BS72]. Applying this to the element s, t s t u t ts st|st|st 2 2 s, t t s u s sts st|st|st 3 1 s, t s, t s, t s, t − Figure 2. All 6 elements in A 2 with their st-sorting word, length, reflection length, left descents, right descents, covered reflections, and covering reflections.
s, t sts · st st|st|st s, t s, t sts · ts st|st|st s, t s, t sts · sts st|st|st s, t s, t u = sw yields for t ∈ des L (sw), that s ∨ t is a left factor of sw. If t = s (and thus ℓ(s ∨ t) > 1), we conclude that tst · · · ℓ(s∨t)−1 factors is a left factor of w. The meet of w 1 , w 2 , . . . , w q is denoted gcd(w 1 , w 2 , . . . , w q ), while their join is lcm(w 1 , w 2 , . . . , w q ). Observe that the Coxeter group W injects into B + (W ) as the interval [e, w • ] Weak(B + ) with w • = lcm(S). This injection preserves the weak order, The embedding of the weak order on W into the weak order on B + suggests the following generalization, which plays a crucial role in the development of the m-eralized theory. It was previously considered in [Deh07].   Figure 4 illustrates the Hasse diagram of Weak (2) (A 2 ).
Each element w ∈ B + has a normal form called Garside factorization garside(w), which we will use to give an alternative description of the m-weak order. To this end, set w 1 = w and v 1 = gcd(w 1 , w • ). For i = 2, 3, . . ., as long as w i−1 = e, let  where w (i) := v i and the degree deg(w) is defined to be k. By construction, every factor sits inside the interval [e, w • ] and so can be treated as an element of W . A factorization v 1 · v 2 · · · · · v k with v i ∈ W is the Garside factorization of the element w = v 1 · · · v k ∈ B + if and only if Proof. We argue by induction on ℓ(u), the base case when u = e being trivial. Otherwise, let u = u ′ s and w ′ = sw for some s ∈ S, so that We have the following proposition, generalizing [EM94, Theorem 2.11], where it was stated for the positive braid monoid, of type A n . Proof. Suppose that w ≤ w r • with r minimal and that w has degree k, so that garside(w) = w (1) · w (2) · · · · · w (k) . We will show that r = k by induction on the number of factors of w.
On the other hand, it is easy to show that k ≥ r by showing that w ≤ w k • : let u 1 be the element such that u 1 w (1) = w • . Push this copy of w • to the right to obtain the factorization (w (2) ) w• · (w (3) ) w• · · · · · (w (k) ) w• · w • . Iterating, we obtain the desired conclusion.
Therefore, if w has degree k, then w ≤ w k • with k minimal, from which we conclude the theorem.
Unfortunately, we do not know of any similar expression for Weak (m) (W ) when 1 < m < ∞-for example, Weak (2) (A 3 ) has 211 elements and the rank generating function is an irreducible polynomial over R.
To prove this theorem, we recall some standard results from the weak order of Coxeter theory in the context of Weak (m) (W ). Just as Weak(W ) has an antiautomorphism given by acting by the longest element w • , we show that Weak (m) (W ) has an anti-automorphism given by acting by w m Lemma 2.28. The map φ(w) := w m • w −1 is an anti-isomorphism from right to left weak order when restricted to Weak (m) (W ).
Proof. We have seen in Lemma 2.27 that w ∈ W (m) is both a left and right divisor of w m • . We also have To show that the map reverses the order, it suffices to consider the case for w ≤ ws ∈ W (m) . Then To convert back from left weak order to right weak order, we introduce the reverse map rev. If u = s i1 · · · s ip ∈ B + (W ), then rev(u) is given by s ip · · · s i1 ∈ B + (W ).
Lemma 2.29. The map rev is an isomorphism from left weak order to right weak order that preserves the interval Weak (m) (W ).
Proof. Since rev(sw) = rev(w)s, rev is an isomorphism from left weak order to right weak order. If w ≤ w m • , then there exists u ∈ B + (W ) such that uw = w m • so that rev(w) rev(u) = rev(uw) = rev(w m Proof. This follows from Lemmas 2.28 and 2.29. Proof of Theorem 2.26. Any interval in a lattice is again a lattice, thus Weak (m) (W ) is a lattice by Theorem 2.20. The self-duality is a direct consequence of the existence of the anti-isomorphism rev •φ. Finally, it is rank-symmetric since it is graded by the Coxeter length.
2.9. Parabolic subgroups and parabolic quotients. For a Coxeter system (W, S) and a subset J ⊆ S, the parabolic subgroup W J is the subgroup of W generated by J, and the corresponding parabolic root subsystem is given by Φ J = Φ ∩ R{α s : s ∈ J}. Likewise, for an Artin system (B, S), the parabolic subgroup B J is the subgroup of B generated by J. The positive parabolic submonoid is denoted . A parabolic subsystem is called maximal if it is generated by all but one generator. We use the notation s := S \ {s} for such situations.
The corresponding parabolic quotients in W and in B + are given as The parabolic quotient of the m-weak order is We continue our development of Coxeter-theoretic results by extending the wellknown decomposition w = w J w J for w ∈ W , w J ∈ W J , and w J ∈ W J to the m-weak order. See also [DMR07, Lemma-Definition 2.1.5].
Proposition 2.31. Fix a subset J ⊆ S. Every w ∈ W (m) has a unique factor- . We need to show that w J has no initial s, for any s ∈ J. Suppose it did, so that , so that by Proposition 2.24, w J s has degree greater than m in W J , and therefore also in W . But then arising from the factorization w = w J w J of Proposition 2.31, and observe that w J ≤ w in Weak (m) (W ).
Remark 2.33. Since any element w ∈ B + lies in some W (m) , for J ⊆ S, we remark that there is a well-defined parabolic decomposition according to J in B + . The map w → w J is again a lattice homomorphism.
Remark 2.34. Parabolic quotients are not well-behaved in W (m) . Although the minimal coset representatives have a natural (left) order, there is generally not a unique maximal element, unlike in the situation of Weak(W ). Moreover, because of the restriction that elements in Weak (m) (W ) have only m Garside factors, not all cosets have the same size.

m-eralized noncrossing partitions
We first recall the definition of noncrossing partitions and their m-eralization in Sections 3.1 and 3.2. In Section 3.3 we then define the m-eralized Kreweras complement, the m-eralized Cambrian recurrence, and the m-eralized Cambrian rotation. We conclude our discussion in Section 3.4 by constructing the m-eralized Cambrian poset on noncrossing partitions.

Noncrossing partitions.
Similarly to the (right) weak order, the absolute order on W is defined by considering the absolute length rather than the usual Coxeter length.
The c-noncrossing partitions under the induced order is a lattice, which we denote by NCL(W, c) [BW08, Theorem 7.8]. NCL(A 2 , st) is drawn in Figure 10 on page 37.
The following proposition is a direct consequence of the definition.
Proposition 3.2. Conjugation by c is an automorphism of NCL(W, c).
In [ABW07], C. Athanasiadis, T. Brady, and C. Watt used, for a bipartite Coxeter element c, the natural edge labelling of the noncrossing lattice NCL(W, c) by reflections to prove that NCL(W, c) is EL-shellable. This shelling gives a unique factorization of each element w ∈ NC(W, c) into reflections that increase with respect to the root order inv R (w • (c)) [ABW07, Definition 3.1]. This factorization was extended to all (standard) Coxeter elements in [Rea07a].
. Each w ∈ NC(W, c) has a unique reduced R-word w c = r 1 r 2 · · · r p with r i ∈ R and r 1 < c r 2 < c < · · · < c r p . We write r ∈ w for r ∈ {r 1 , . . . , r p } in this case.
Note the pleasant similarity between Proposition 3.3 and the definition of csorting word in Section 2.7.
The Kreweras complement Krew c : In particular, Krew c (w) · w = c. The operation Krew c turns out to be an anti-isomorphism of the lattice NCL(W, c). By pairing a noncrossing partition with its Kreweras complement and using the unique factorization given in Proposition 3.3, we obtain a simple characterization of noncrossing partitions [Arm06, Tza08, BRT12].
Proposition 3.4. There is an explicit bijection between c-noncrossing partitions NC(W, c) and subwords be the pair obtained by applying the unique factorizations from Proposition 3.3 to each of δ 0 and δ 1 , each specifying certain positions in inv R (w • (c)). By construction, this yields a bijection between NC(W, c) and The order imposed on the reflections in this characterization of NC(W, c) provided by Proposition 3.4 will later be essential for our understanding of Cambrian posets on noncrossing partitions in Section 3.4.
D. Armstrong observed that one can also specify m-noncrossing partitions by instead considering the intervals between consecutive elements of such a chain [Arm06, Definition 3.2.2(2), Lemma 3.2.4].
. We denote the set of all such sequences by NC Proof. Immediate from the definitions.
There is more than one way to define the map w ↔ δ w . Our convention is chosen to ease the connection between m-eralized c-noncrossing partitions, m-eralized csortable elements, and the m-eralized c-cluster complex.
The two equalities ℓ R (cδ 0 c −1 ) = ℓ R (δ 0 ), and together with Proposition 3.2, imply that Krew c (δ) is again an m-delta sequence. It is well-known that the square of the Kreweras complement may be interpreted as a rotation on the combinatorial realizations of m-eralized noncrossing partitions in the classical types [Arm06]. In general, we have the following proposition.
Proof. This follows from the definition of the Coxeter number h as the order of c.
Analogously to Proposition 3.4, one can describe m-eralized c-noncrossing partitions. By Lemma 2.18, the two words inv We obtain the following proposition.
Proposition 3.11. There is an explicit bijection between m-eralized c-noncrossing partitions NC (m) (W, c) and NC Proof. This is nearly identical to the proof of Proposition 3.4, except that our m-delta sequence has m + 1 components (rather than just two). As before, the proposition follows by applying the factorization of Proposition 3.3 separately to each component of the m-delta sequence.
We may denote a facet I of NC n ), such that r 1 · · · r n = c and where each reflection r j is colored by which copy of inv R (w • (c)) it lies in, so that 0 ≤ i 1 ≤ · · · ≤ i n ≤ m. We write r ∈ I for r ∈ {r 1 , . . . , r n }.
As we have now simple, explicit bijections between the three sets NC (m) (W, c), NC  ∆ (W, c), we will freely move between the three descriptions.
Example 3.12. Figure 5 shows all 12 elements of We close this section with the definition of the support of m-eralized c-noncrossing partitions.
Definition 3.13. Let w = (w 1 ≥ R · · · ≥ R w m ) ∈ NC (m) (W, c) and let δ w = (δ 0 , . . . , δ m ) be the corresponding element in NC sut.sut.sut s, t Alternatively, one can understand Shift s on the level of individual reflections by defining it on the subwords NC obtained by placing the s at the end while preserving the values of all noninitial reflections. Otherwise, we associate the subword to its image in Sub R (inv R (Q ′ ), c ′ ) under cyclically rotating Q (which has the effect of conjugating all reflections by s).
Then this gives a bijection up to commutations, by Proposition 2.14 we have a second bijection The composition of these two bijections is the map We will see analogous shift operations again in Section 4.3 and Section 5.5. Example 3.14. As before, consider NC (2) (A 2 , st). Then one orbit of the shift operation is given by We may now compose the shift operations in the order specified by any reduced S-word for the Coxeter element c. This composition does not depend on the chosen reduced word since two shifts Shift s and Shift t commute for commuting s, t ∈ S.
for any reduced S-word s 1 s 2 · · · s n for c.
As an example, observe that the elements of NC (m) (A 2 , st) in Figure 5 are grouped into their orbits under Cambrian rotation, the first one being the orbit discussed in Example 3.14.
Rather than append a final s to δ m in the first case of the definition of the shift operator, we could send δ w to the delta sequence , performing a descent into a parabolic subgroup. This modification of Cambrian rotation yields the m-eralized c-Cambrian recurrence on noncrossing partitions, which we phrase as the following characterization of NC Proof. The recurrence follows directly from Lemma 2.18(6) and (7).
Remark 3.17. Some care must be taken to correctly run this recurrence in reverse. For any (δ 0 , δ 1 , . . . , δ m ) ∈ NC  δ (W s , s −1 c). When running the recurrence backwards, it is therefore crucial to not begin with an element of NC We emphasize once and for all here that in order to reverse any of the Cambrian recurrences in Propositions 4.2, 4.9, 4.14, 5.31 and 4.32, we will always require that the elements on the right hand side have the form specified by the recurrence.
Just as with Cambrian rotation, Proposition 3.16 can be refined to a statement on NC ∆ (W, c) under the canonical factorization of Proposition 3.3. .

m-eralized Cambrian lattices.
To place the definition of our m-eralized c-Cambrian lattice in context, we first consider the situation for m = 1.
The weak order Weak(W ) can be described using the covering and covered reflections of the elements in W . Given that cov ↑ (e) = S and cov ↓ (e) = ∅, one can reconstruct the covered and covering reflections of all elements as follows. Supposing that cov ↑ (w) and cov ↓ (w) are known, choose some r ∈ cov ↑ (w) and let α r be the associated positive root. The sets cov ↑ (rw) and cov ↓ (rw) are then given by An element w ∈ W can be reconstructed from its covered and covering reflectionsthese tell us exactly which hyperplanes bound the corresponding Weyl chamber, which determines w up to multiplication by w • . Since we have distinguished covering and covered reflections, we have therefore uniquely specified w.
Remark 3.19. One naive m-eralization of the weak order would be to allow m + 1 sets of reflections (cov 0 , cov 1 , . . . , cov m ), generalizing the two sets cov ↑ and cov ↓ . Setting the minimal element to be ({s 1 , s 2 , · · · s n }, ∅, . . . , ∅), we were unable to find a m-eralization of the m = 1 action by (simple) reflections for which there was a unique maximal element in this framework.
We will next mimic the characterization of Weak(W ) given above the remark.
The analogy we wish to draw is that the factorization of δ 0 of Proposition 3.3 should be thought of as a set cov ↑ (w), while the factorization of δ 1 behaves like cov ↓ (w). Jumping immediately to the definition for general m does not introduce any additional complication.
Let I be a facet of NC (m) ∆ (W, c), and consider I as a reduced R-word r 1 · · · r n for c as a subword of Observe that for any reflection r ∈ R, every other reflection appears exactly once between two consecutive copies of r inside Q. If r = r j does not appear inside the final copy of w • (c), the increasing flip Flip ↑ r (I) is given by r 1 · · · r j−1 r r j+1 · · · r r k r j r k+1 · · · r n , where k is chosen maximally such that r k still appears before the next consecutive copy of r inside Q, and the reflections chosen between those two consecutive copies of r are r r j+1 , . . . , r r k . By Proposition 3.20 below, this is again a subword of Q. We define the decreasing flip Flip ↓ r (I) for r not appearing in the initial copy of w • (c) analogously.
Just as for covered and covering reflections in Weak(W ), individual reflections are shuttled between the two copies of inv(w • (c)) in Q to maintain the structure imposed by the absolute order on R induced by c. Examples of increasing and decreasing flips can be found in Figure 6.
We must check that the reflections r r j+1 · · · r r k indeed appear in this order between two copies of r inside Q. We have already mentioned that every reflection except r appears exactly once between these two copies of r. Denote by Q r this sequence, together with an initial r. Let w • (c) = s 1 . . . s N and let the element w ∈ W be given by the prefix s 1 · · · s i−1 such that r = s w i .   to w • (c) along the sequence s 1 , . . . , s i−1 yields that s i is initial in c ′ with c = (c ′ ) w , where Q r is the sequence given by t w for t ∈ inv R (w • (c ′ )). The sequence r w −1 j · · · r w −1 k is contained in NC(W, c ′ ) starting with r w −1 j = s i . It follows (see e.g. [BDSW14, Theorem 1.4]) that each of r w −1 j+1 , . . . , r w −1 k lives in the parabolic subgroup W si . By Lemma 2.18(7), the ordering of the reflections in W si agrees in each of w • (c ′ ), w • (s −1 i c ′ ), and w • (s −1 i c ′ s i ), so that conjugating the reflections r w −1 j+1 , . . . , r w −1 k by s i does not change their order. Conjugating this statement by w, we conclude that the word r r j+1 · · · r r k appears in the same order in Q r as the word r j+1 · · · r k .
In analogy to the weak order, flips can be used to define an m-eralized c-Cambrian poset on NC (m) (W, c).   We may also define a Cambrian graph, where we allow longer flips that send a reflection from the i-th copy of w • (c) to the j-th copy for i < j.    Proposition 3.23. We have for c = s 1 · · · s n and c ′ = s w• n · · · s w• 1 that (i) the posets Camb The description in terms of m-delta sequences follows from the observation that the absolute lengths ℓ R (δ 0 · · · δ m−1 ) and ℓ R (δ 1 · · · δ m ) are equidistributed on all m-delta sequences in NC if s ∈ δ 0 and r = s; Proof. This follows immediately from the definition of Shift s in Section 3.3 and the definition of Flip ↑ r . Corollary 3.25. Let c, c ′ be two Coxeter elements. Then there is an undirected graph isomorphism GCamb

m-eralized Coxeter-sortable elements
In this section, we define an m-eralization of c-sortable elements as a certain subset Sort (m) (W, c) of W (m) . We start by reviewing N. Reading's theory of c-sortable elements [Rea06,Rea07b,Rea07a], recalling two characterizations of these elements in Definition 4.1 and Proposition 4.2. We m-eralize c-sortable elements in Section 4.2, providing the analogous characterizations as Definition 4.5 and Proposition 4.9. We give a characterization of Sort (m) (W, c) on Garside factors in Definition 4.13, and we give a bijection between Sort (m) (W, c) and noncrossing partitions in Section 4.5. Generalizing the case for m = 1, we prove in Section 4.6 that Sort (m) (W, c) is a sublattice of Weak (m) (W ). Finally, in Section 4.7, we explain the connection between Sort (m) (W, c) and NC (m) (W, c) by giving a bijection between Sort (m) (W, c) and chains in N. Reading's shard intersection order. 4.1. Coxeter-sortable elements. N. Reading introduced and studied c-sortable elements in [Rea06,Rea07a]. The c-sortable elements have three different characterizations, each of which is useful in different ways.
Definition 4.1 (N. Reading [Rea06]). An element w ∈ W is c-sortable if the csorting word w(c) for w defines a decreasing sequence of subsets of positions in c. We denote the set of c-sortable elements by Sort (W, c). The c-Cambrian lattice Camb Sort (W, c) is the restriction of Weak(W ) to Sort(W, c).
Although the definition of being c-sortable depends on a particular choice of a reduced word c for the Coxeter element c, we have seen in Section 2.7 that all c-sorting words w(c) are commutation equivalent. Therefore, the property of being c-sortable does not depend on a particular chosen word.
The second characterization is the c-Cambrian recurrence, which is immediate from Definition 4.1.
N. Reading's third characterization describes c-sortable elements by their inversion sets as the c-aligned elements. We do not m-eralize this definition, and so omit further discussion, except to summarize the following properties of c-sortable elements. (1) w J ∈ Sort(W J , c J ), where J ⊆ S and c J is obtained from c by deleting the letters S − J from any reduced word for c;    It is not c-sortable since s 1 occurs in the fourth but not the third copy of c. Note that the vertical bars here serve only to distinguish the copies of c.
As in the situation of m-eralized c-noncrossing partitions, we conclude this section by discussing the support of m-eralized c-sortable elements. Since these are elements in the positive Artin monoid B + (W ), they inherit the notion of support from Section 2.2. Figure 7 shows the support of the 12 elements in Sort (m) (A 2 , st).

m-eralized
Cambrian rotation and recurrence. The Cambrian rotation and the Cambrian recurrence both depend on an operation Shift s for an initial simple reflection s in a Coxeter element c. The map This operation does not a priori yield an element in Sort (m) (W, s −1 cs). The proof that w ∨ s m is indeed (s −1 cs)-sortable is will be given in Proposition 4.25.
Example 4.7. Parallel to Example 3.14, we consider Sort (2) (A 2 , st). One orbit of the shift operation is given by After removing the initial e, which is thought of as an element of Sort (2) (A 2 , st), the right column consists of elements of Sort (2) (A 2 , st), while the left column contains elements of Sort (2) (A 2 , ts).
We may now compose the shift operations in the order specified by any reduced S-word for the Coxeter element c. This composition does not depend on the chosen reduced word, since two shifts Shift s and Shift t commute for commuting s, t ∈ S. for any reduced S-word s 1 s 2 · · · s n for c.
Rather than take the join with s m in the first case of the definition of the shift operator, we could have sent w to itself, viewed as an element of a parabolic subgroup. We call this process, as given in the following proposition, the m-eralized c-Cambrian recurrence on Coxeter-sortable elements.
Proposition 4.9. Let s be initial in a Coxeter element c. Then   shard (A 2 , st) will be defined and studied in Section 4.7.  Let w ∈ W , let c = s 1 s 2 · · · s n be a Coxeter element, and let des R (w) = {s i1 , s i2 , . . . , s i k }, where s w i1 ≤ c s w i2 ≤ c · · · ≤ c s w i k in the root order associated to c. Define the restriction of c with respect to the element w to be the Coxeter element c w := s i1 s i2 · · · s i k of the Coxeter system given by the standard parabolic subgroup W desR(w) generated by the simple reflections in des R (w). Observe that c w contains the same simple reflections as the restriction of c to this standard parabolic subgroup, but that the order in which the simple reflections appear is not necessarily the order in which they appear in c-rather, it is the order coming from the root order associated to c. Definition 4.13. Let w be an element in W (m) with garside(w) = w (1) · · · · ·w (m) . We say that w is factorwise c . We denote the set of factorwise c-sortable elements by Sort      if s ∈ des L (w) .
We consider both implications individually in the final case su (1) ≤ w • . Note that in this case, s ∈ des L (u (1) ).
Suppose u ∈ Sort (m) fact (W, c ′ ). Since s is the final reflection in the reflection order associated to c ′ = s −1 cs, the simple reflection s is the final reflection in the inversion sequence inv(u (1) (c ′ )) (since u (1) is c ′ -sortable), and so corresponds to the final simple reflection r in the c ′ -sorting word u (1) (c ′ ). More succinctly, we can write u (1) r = su (1) , so that u (1) = su (1) r −1 . In this case, set w (1) = u (1) = su (1) r −1 , which is c-sortable by Definition 4.5.
We drop the first Garside factor and repeat the preceding argument on the element ru (2) · · · u (m) , obtaining an element of Sort We claim that w (1) · w (2) · · · · · w (m) is the Garside factorization of w. Since we did not change the first Garside factor w (1) = u (1) , since r ∈ des R (u (1) ) and des L (ru (2) ) ⊆ {r} ∪ des L (u (2) ) (by Proposition 2.19), w (2) lies in W u (1) . We conclude that w (1) was indeed the first Garside factor of w. The result follows by induction on the number of Garside factors of w. fact (W, c). Running the argument above in reverse, we obtain the candidate Garside factorization of u, u (1) = w (1) and u (2) · · · · · u (m) = r −1 w (2) · · · w (m) , where u (1) r = su (1) . Since des R (u (1) ) = des R (w (1) ), and since w (2) only uses simple reflections in des R (w (1) ), u (1) is indeed the first Garside factor. The result again follows by induction on the number of Garside factors of w.
Proof. Let w ∈ Sort (m) (W, c) with s initial in c and s ∈ des L (w). If we let w = su, then the proof of Proposition 4.14 shows that the Garside factorization of u is obtained by removing the initial s from the Garside factorization of w and performing commutations. The same relationship is trivially true for the sorting words of w and u. The result now follows from the Cambrian recurrence.  Proof. Let w = w (1) · · · · · w (m) be the Garside factorization of w. The factor w (i) lives, by Definition 4.13, inside the parabolic generated by des R (w (i−1) ). Note that the longest element of the parabolic subgroup generated by des R (w (i−1) ) is a right factor of w (i−1) . The statement then follows by Proposition 4.17 and induction.
The next example shows that this property does not hold in general for nonsortable elements.  Proof. The inversion set inv(w J ) is contained in the restriction of inv(w), since it it an initial segment of w. We now show that there are exactly the right number of inversions in inv(w) ∩ Φ + (W J ). Consider the roots in W J coming from the ith Garside factor of w; by restricting to w J , we compute from Proposition 2.32 that the roots in the ith Garside factor of w J are w (1) In particular, there are the same number of roots in W J in the ith Garside factor of w as there are in w J , and so inv(w J ) = inv(w) ∩ Φ(W J ).
The second statement follows directly from Proposition 2.32 and the statement for m = 1, given in  (1) w ≤ u if and only if inv(w) ⊆ inv(u); and (2) inv(w ∧ u) = inv(w) ∩ inv(u).
Proof. We prove w ≤ u if and only if inv(w) ⊆ inv(u) first. If w ≤ u then it is clear that inv(w) is contained in inv(u), since w is a left factor of u. We now argue the converse. Suppose inv(w) ⊆ inv(u).
• If s ∈ asc L (w) and s ∈ asc L (u), then we are done by restriction to W s .
• We cannot have s initial in w but not in u, so suppose that s ∈ asc L (w) and s ∈ des L (u). It is clear that u s ≤ u. Since w ∈ Sort (m) (W s , s −1 c), we have that inv(w) ⊆ inv(u s ) by Proposition 4.21, so that by induction on rank (since both w, u s ∈ Sort (m) (W s , s −1 c)), w ≤ u s . Since u s ≤ u, we conclude that w ≤ u. • Finally, if s ∈ des L (w) and s ∈ des L (u), then we get the statement for s −1 u = u ′ and s −1 w = w ′ by induction on length. Multiplying by s then does not change containment of inversion sets (since multiplication by s just conjugates all reflections by s, and then adds s). We now show inv(w ∧ u) = inv(w) ∩ inv(u).
• If s ∈ asc L (w) and s ∈ asc L (u), then we are done by restriction to W s .
• If s ∈ des L (w) and s ∈ asc L (u), then u ∈ Sort (m) (W s , s −1 c) so that w ∧ u = (w ∧ u) s = w s ∧ u. We conclude the result by induction on rank. • The case s ∈ des L (u) and s ∈ asc L (w) follows by symmetry.
• Finally, if s ∈ des L (w) and s ∈ des L (u), then we obtain inv( Remark 4.23. The most naive guess for how weak order might be characterized on Sort (m) (W, c) in a componentwise fashion would be to compare individual Garside factors in weak order-this naive guess is wrong. For example, s · s and sts · s are not comparable, even though individually s ≤ sts and s ≤ s.
Remark 4.24. We emphasize that the first part of Theorem 4.22 does not hold in general for non-sortable elements, although it is true when m = 1. The second part doesn't even hold when m = 1 for non-sortable elements.
The remainder of this section is devoted to the following technical proposition, which we used previously to establish that the shift operator Shift s : Proposition 4.25. Let c be a Coxeter element, let s be initial in c, and let w ∈ Sort (m) (W s , sc). For 0 ≤ k ≤ m, w ∨ s k is simultaneously c-sortable and (s −1 cs)sortable.
Let w(0) := w (1) 0 · · · · · w (m) 0 be the Garside factorization of the element w in the proposition, and then inductively set where s k and v k are the following elements of W : We will show that the decomposition w(k) = w (1) k · · · · · w (m) k is the Garside factorization of w ∨ s k , and that this factorization is factorwise c-and (s −1 cs)sortable.
Lemma 4.26. The element w(k) is cand (s −1 cs)-sortable with Garside factorization w As the proof of this lemma is a slightly involved induction on k, we extract the base case k = 1 into a separate lemma for readability.
Lemma 4.27. The element w(1) is cand (s −1 cs)-sortable with Garside factorization Proof. Since w(0) is c-sortable (and thus factorwise c-sortable) by assumption, its first factor w The factorwise c-sortability also implies that the Garside factors w Proof of Lemma 4.26. We write c (i) j for the Coxeter element c (i) for w(j), as defined at the beginning of Section 4.4. We will prove the statement of Lemma 4.26 along with by induction on k. These are established for k = 1 by Lemma 4.27. It remains to conclude the statements for k, assuming that they hold for k − 1.
The first k − 1 Garside factors have not changed, and so are still Garside factors, and each of them is sortable in its corresponding parabolic subgroup given by the definition of factorwise sortability.
We next show that s k ∈ des R (w (k−1) k−1 ) (which, in particular, shows that it is a simple reflection) and s k / ∈ supp(w (k) k−1 ). We can assume by induction that k−1 corresponding to its cover reflection s k−1 , implying the first property s k ∈ des R (w in the same way as w The induction hypothesis gives us that s k−1 is initial in c k−1 ) is the cover reflection corresponding to s k ∈ des R (w (k−1) k−1 ) and s k−1 . We can therefore apply Lemma 4.4 to the c k−1 lives in the parabolic subgroup generated by cov ↓ (w The final part of the proof is to conjugate the remaining Garside factors w by v k . This part is completely analogous to the argument given in the proof of Lemma 4.27. Proof of Proposition 4.25. We show that w(k) = w ∨ s k , and again first consider the case k = 1.
Clearly, s ≤ w(1) since the Garside factorization begins with w (1) 0 ∨ s which is above s in W and therefore has a reduced word starting with s. Also w ≤ w(1) since as before, and write v −1 The first equality is given by the definition of w(1) in terms of w(0). Then Lemma 4.27 implies that w (1) 1 · · · · ·w (m) 1 is indeed the Garside factorization w(1).
It remains to show that w(1) is minimal among all elements above s and w. Although the colored inversion set of an element of B + is not necessarily unique to that element, the number of inversions still tell us its length. Any element above w must contain all inversions of w, and any element above w (1) 0 and s must contain the inversions of w (1) 0 ∨ s. The inversion set of w(1) contains all these inversions and no others, and therefore has the minimal desired length; we conclude that w(1) = w ∨ s.
For the case of general k, we first check that w(k −1) ≤ w(k) and that s k ≤ w(k). For w(k − 1), we have For s k , we have We now show that w(k) is the minimal element above s k and w(k − 1). Let u = (w (1) k−1 ). We claim that s k ∨ w(k − 1) = (su) ∨ w(k − 1). We first show that s k ∨ u = su. The element s k ∨ u is divisible by s k , and since u is divisible by s k−1 , the length of s k ∨ u is at least one more than the length of u. As the element su = us k is divisible by both s k and u, we conclude that su = s k ∨ u. Therefore, since u is a left factor of w(k − 1), we conclude that Now su = us k , so that us k and w(k − 1) share their first k − 1 Garside factors. Therefore, k−1 )). By the inversion set argument used above for k = 1, we may now conclude that the final m − k + 1 Garside factors s k ∨ (w k k−1 · · · · · w (m) k−1 )) are of specified form, so that w(k) = s k ∨ w(k − 1).
Example 4.28. Proposition 4.25 does not hold for non-sortable elements. For example, in type A 3 , if c = s 1 s 2 s 3 and u = s 3 s 3 s 2 , then u ∨ s 1 = s 3 s 3 s 2 s 1 s 2 = s 3 s 3 s 1 s 2 s 1 = s 1 s 3 s 3 s 2 s 1 = s 1 s 3 · s 3 s 2 s 1 .

m-eralized
Coxeter-sortable elements and noncrossing partitions. We begin this section with a direct construction of a bijection between m-eralized csortable elements an m-eralized c-noncrossing partitions.
Generalizing the situation in [RS11] and in [PS11,Proposition 6.20], we define the skip set for c-sortable elements as follows. Given w ∈ Sort (m) (W, c), let its csorting word be w(c) = s 1 · · · s p and let s ∈ S. We say that w skips s in position k+1 if the leftmost instance of s in c ∞ not used in w(c) occurs between s k and s k+1 . The skip set (of colored positive roots) is then defined by is the colored positive root s 1 · · · s k (α (0) s ) and k is chosen such that w skips t in position k + 1. A skip is a-forced if it occurs with color a; a skip is forced if it is m-forced and unforced otherwise. As in the definition of root configurations of facets of subword complexes, we consider skip sets as naturally ordered induced by the order of S induced by c. (As usual, this ordering is only defined up to commutations.) Again, whenever we write a union of colored roots in a skip set, we mean the disjoint union ordered by concatenation of the individual orderings.
Remark 4.29. This terminology comes from the following observation: for w ∈ Sort (m) (W, c), if w is modified to postpone an m-forced skip, then the resulting element is no longer less than or equal to w m • .
Example 4.30. Figure 7 shows the skip sets of all elements in Sort (2) (A 2 , st).
Remark 4.31. It is easy to recover the c-sorting word w(c) for w ∈ Sort (m) (W, c) from its skip set C c (w) as follows. Begin reading the word c ∞ = s 1 s 2 . . . from left to right; we will specify a procedure for deleting letters. If there is no next letter, the letters remaining spell w(c). Otherwise, if the next letter in c ∞ is a skip-in the sense that the corresponding colored positive root is contained in C c (u), where u is the product of the undeleted letters strictly left of the current position-then it and all of its occurrences to the right in c ∞ are deleted from c ∞ .
The skip set C c (w) is closely linked to NC Proof. If s ∈ asc L (w) then w ∈ Sort (m) (W s , s −1 c) with skip set C s −1 c (w). If we treat w as an element of Sort (m) (W, c), this does not change the positions of the skips t = s. But w now skips s in position 1, so that the new colored positive root α (0) s is added to the skip set. The second case for s ∈ des L (w) is similar-no simple reflection is skipped in position 1, and so each colored positive root s 1 · · · s k (α t ) ∈ C c (w) corresponds to a colored positive root s 2 · · · s k (α t ) ∈ sC s −1 cs (s −1 w).
Example 4.33. Considering sts · s ∈ Sort (2) (A 2 , st) as in Example 4.10, the sequence of skip sets is , the maximal possible color in C c (w) is indeed m. We conclude the following theorem.     4.6. m-eralized Cambrian lattices. N. Reading's c-sortable elements are the key to understand certain order congruences on the weak order that respect the lattice structure of the weak order (and are therefore lattice congruences). We briefly summarize some results of [Rea07b]. N. Reading defined an order-preserving projection π c ↓ : Weak(W ) → Sort(W, c) sending an element w to the largest c-sortable element less than or equal to w. Likewise, there is a related order-preserving map π ↑ c that maps w to the smallest c-sortable greater than or equal to w. N. Reading showed that the fibers of π c ↓ and π ↑ c are equal and that the fiber containing w is the interval [π c ↓ (w), π ↑ c (w)] Weak(W ) . This turns out to be enough to conclude that the c-sortable elements form a lattice quotient of the weak order. Each of these congruences defines an associahedron corresponding to c; the 1-skeletons of these c-associahedra are the Hasse diagrams of the c-Cambrian lattices. In contrast, the m-eralized c-sortable elements no longer form a lattice quotient of Weak (m) (W ), as indicated in the following example.
Example 4.37. Let π c ↓ : Weak (m) (W ) → Sort (m) (W, c) be defined as the largest c-sortable element less than or equal to w. In type A 2 with m = 3, this does not define a lattice quotient, as we now illustrate. The element s 2 s 1 ·s 1 ·s 1 s 2 is maximal in the fiber above s 2 because its one cover is the element w • · s 1 s 2 · s 2 , which is above w • · s 1 . On the other hand, the element s 2 s 1 · s 1 s 2 · s 2 s 1 is also maximal in the fiber above s 2 because its single cover is the element w • · s 1 s 2 · s 2 s 1 , which is also above w • · s 1 . Therefore, there is more than one maximal element in the fiber above s 2 , the fiber is not an interval, and so π c ↓ cannot define a lattice quotient. One might hope that the failure arose because we were "missing" the full fiber above s 2 by restricting to only those elements below w 3 • . It would make sense to therefore take the entire monoid B + to obtain the fiber. In fact, this does not resolve the problem, since (s 2 s 1 · s 1 · s 1 s 2 ∨ s 2 s 1 · s 1 s 2 · s 2 s 1 ) = w • · s 1 s 2 · s 2 s 1 ≤ w 3 • .  Remark 4.39. F. Bergeron defined an m-Tamari lattice on m-Dyck paths. When m = 1, this lattice is isomorphic to Camb Sort (A n , s 1 · · · s n ), but this is no longer the case for m > 1. The m-Tamari lattice is shown for m = 2 in Figure 8; compare with Figure 9. We refer to [Ber12] for definitions and further details. For conjectural similarities between the two lattices, see Remark 5.44.
Although we no longer have a lattice quotient, the restriction of Weak (m) (W ) to Sort (m) (W, c) still yields a lattice.  Proof. The proof is analogous to the proof of [Rea07b, Theorem 1.2], although-as illustrated in Example 4.37-we do not have a projection map π c ↓ , and so cannot rely on its properties to compute the join.
We first show that u ∧ v ∈ Sort (m) (W, c) for u, v ∈ Sort (m) (W, c). Let s be initial in c.
• If s ∈ asc L (u) and s ∈ asc L (v), then u, v ∈ Sort (m) (W s ) and we conclude the result by induction on rank. • If s ∈ des L (u) and s ∈ asc L (v), then v ∈ Sort (m) (W s , s −1 c). Since w → w s is a meet-semilattice homomorphism by Proposition 2.32, By Proposition 4.21, u s is (s −1 c)-sortable, so that u ∧ v is c-sortable by the previous case. • The case s ∈ des L (v) and s ∈ asc L (u) follows by symmetry.
• Finally, suppose s ∈ des L (u) and s ∈ des L (v). Let c ′ = s −1 cs so that s −1 u and s −1 v are both c ′ -sortable. By induction on length, , which is c-sortable by Proposition 4.9.
We now show that u ∨ v ∈ Sort (m) (W, c) for u, v ∈ Sort (m) (W, c). Again, let s initial in c.
• If s ∈ des L (u) and s ∈ des L (v), then s ∈ des L (u ∨ v). Let c ′ = s −1 cs so that s −1 u and s −1 v are both c ′ -sortable. By induction on length, , which is c-sortable by Proposition 4.9. • If s ∈ des L (v) and s ∈ asc L (u), then u ∈ Sort (m) (W s , s −1 c) and s ∨ u is csortable by Proposition 4.25. We compute that u∨v = s∨(u∨v) = (s∨u)∨v, so that u ∨ v is c-sortable by the previous case. • The case s ∈ des L (u) and s ∈ asc L (v) follows by symmetry.
• Finally, suppose s ∈ asc L (u) and s ∈ asc L (v). Then u, v ∈ Sort (m) (W s ) and we conclude the result by induction on rank.
Example 4.41. Figure 9 shows all 12 st-sorting elements in Camb In the remainder of this section, we prove that the two posets Camb      NC (W, c) is a lattice. We will prove this theorem by induction, using the Cambrian recurrence. As a preliminary result, we analyze the situation of the skip set in Proposition 4.25. Let c be a Coxeter element, let s be initial in c, and let w ∈ Sort (m) (W s , sc). For 0 ≤ k ≤ m, we have seen that w ∨ s k is c-sortable and (s −1 cs)-sortable.  Proof. The proof of Proposition 4.25 describes exactly how the Garside factorization of w∨s k−1 is obtained from the Garside factorization of w. This shows that the colored inversion sequence of the c-sorting word only changes within the (k − 1) st Garside factor. Since we know from the case m = 1 that the roots in this Garside factor change exactly in the described way [RS11, Proposition 5.4], we conclude that the skip set C c (w ∨ s k ) is obtained from the skip set C c (w ∨ s k−1 ) by e s t st sts sts · s sts · t sts · ts sts · sts s · s t · t st · t  The lemma then follows by applying this procedure k times, starting with w = w ∨ s 0 .
Proof of Theorem 4.42. We show that a cover relation u ⋖ v in Camb Sort (W, c) corresponds to a cover relation I ⋖ J in Camb NC (W, c) under the map given in Theorem 4.34. To this end, let δ = (δ 0 , . . . , δ m ) and δ ′ = (δ ′ 0 , . . . , δ ′ m ) be the delta sequences corresponding to I and to J, respectively.
• If s ∈ asc L (u) and s ∈ asc L (v) (equivalently, s ∈ δ 0 and s ∈ δ ′ 0 ), the statement follows by the Cambrian recurrences given in Propositions 4.2 and 3.16 and their relation in Theorem 4.34.
• The case s ∈ des L (u) and s ∈ asc L (v) (equivalently, s / ∈ δ 0 and s ∈ δ ′ 0 ) is impossible since u ≤ v in weak order.
• Finally consider the case s ∈ asc L (u), s ∈ des L (v) (equivalently, s ∈ δ 0 , s / ∈ δ ′ 0 ). We first analyze Flip ↑ r (I) = J. By Equation (5), the set of (m + 1)colored positive roots in J is obtained from the set of (m + 1)-colored positive roots in I by replacing α , and leaving all roots of the form β (ℓ) for ℓ > 1 unchanged. On the other hand, we have that u ≤ v in weak order and, since s ∈ supp(v), s ≤ v in weak order, implying that u ∨ s ≤ v in weak order. Since s ≤ vs / ∈ supp(u), we can apply Lemma 4.43 with k = 1 and obtain that u ∨ s is c-sortable. Therefore, v = u ∨ s and the skip set of v is obtained from the skip set of u in the expected way.   In Theorem 4.52, we give a bijection between this seemingly artificial m-eralization of c-sortable elements as chains and Sort (m) (W, c). As a corollary, we obtain a second bijection between Sort (m) (W, c) and NC (m) (W, c); we show in Theorem 4.56 that this recovers the bijection previously given in Theorem 4.34.
In [Rea11], N. Reading defined a delicate slicing procedure on simplicial hyperplane arrangements that cuts hyperplanes into several pieces called shards. The shard intersection order Shard(W ) is the set of all intersections of these hyperplane pieces, ordered by reverse inclusion. N. Reading proved that the intersection of the lower shards of an element w ∈ W is a bijection between W and the set of shard intersections. The longest element w • is mapped to the maximal element in the shard intersection order under this bijection. It is indeed possible to define the shard intersection order directly on W . Example 4.45. Figure 10 shows the two lattices NCL(A 2 , st) and Shard(A 2 ) restricted to Sort(A 2 , st). Figure 11 shows Shard(A 3 ) restricted to Sort(A 3 , s 1 s 2 s 3 ). In both figures, the sortable elements are given by their inversion sets in the positive roots. Covered reflections are circled in grey, and further inversions in the parabolic subgroup generated by the covered reflections are circled in white.
A (non-stuttering) gallery, or simply gallery, is a walk on the connected regions of the complement of the hyperplane arrangement of W , where two regions are connected if they share a bounding hyperplane, and any hyperplane is crossed at most once.  Proof. Let H be the hyperplane corresponding to a lower shard t of v and consider a different shard t ′ in H. It is not hard to see from the definition of shard [Rea11] that there is a hyperplane K which cuts H such that t and e are on one side of K, and t ′ is on the other side. Then it is clear that a gallery from e to v will not cross K and so will not cross t ′ . Proof. We first show that cov ↓ (u) ⊆ cov ↓ (v) and inv(u) ⊆ inv(v) implies u v. By the correspondence between the lattice of (conjugates of) parabolic subgroups and the intersection lattice of the hyperplane arrangement of W [BI99], cov ↓ (u) ⊆ cov ↓ (v) implies that H.
If we in addition have inv(u) ⊆ inv(v), then there exists a gallery from e to u to v. Let ‫שׁ‬ be the union of the set of shards that this gallery crosses, and note that because the gallery is non-stuttering, there is exactly one shard for each hyperplane in inv(v). For any region r with lower hyperplane H, the entire facet of r corresponding to H is part of the same shard. By Lemma 4.46, each hyperplane in cov ↓ (r) intersected with the shards arising from a gallery gives the corresponding lower shard of r. Now taking the intersection of both sides of Equation (7) with ‫שׁ‬ implies that the intersection of the lower shards of u is contained in the intersection of the lower shards of v.
We now show that u v implies that cov ↓ (u) ⊆ cov ↓ (v) and inv(u) ⊆ inv(v). By [Rea11, Proposition 5.5], we know that cov ↓ (u) ⊆ cov ↓ (v) , since sending shards to their containing hyperplane induces an order-preserving map between the shard intersection order and the intersection lattice of W . Finally, by [Rea11, Proposition 4.7 (ii)], sending regions to the intersection of their lower shards is an order-preserving map between the shard intersection order and the weak order.  Proposition 1.2]). The interval [e, w] Shard(W ) is isomorphic to Shard(W cov ↓ (w) ) ∼ = Shard(W desR(w) ).
The isomorphism [e, w] Shard(W ) ∼ = Shard(W cov↓(w) ) is given by sending u w to u cov↓(w) , the element in W cov ↓ (w) with inversion set inv(u)∩Φ + (W cov ↓ (w) ). Likewise, the isomorphism [e, w] Shard(W ) ∼ = Shard(W des R (w) ) is given by sending u w to u desw , the element in W desR(w) with inversion set N. Reading showed that the shard intersection order provides an alternative way of thinking about the noncrossing partition lattice NCL(W, c): it is the restriction of Shard(W ) to the sortable elements Sort(W, c).         This element is evidently not c-sortable.
Proof of Theorem 4.52. Given a multichain (w 1 w 2 · · · w m ) ∈ Sort (m) shard (W, c), we produce an element in Sort (m) (W, c) as follows. Recall that Proposition 4.48 gives a bijection between the interval [e, w] Shard(W ) and Shard(W des R (w) ). It follows from the bijection between c-sortable elements and noncrossing partitions in [Rea11] that if w u with w, u ∈ Sort(W, c), then u is sent to a c w -sortable element in W desR(w) . This process is bijective, since every step is bijective.
The inverse of this bijection is given by explicitly describing the inverse of Proposition 4.48 on c-sortable elements: given w ∈ W des R (w) , we conjugate it by w to an element of W cov ↓ (w) . Since c-sortable elements are uniquely defined by their cover reflections, there is now a unique way to complete the inversion set in W cov↓(w) to the inversion set of a c-sortable element in W by filling in initial segments of dihedral subgroups of W according to their c-orientation.
Remark 4.54. In analogy to componentwise absolute order on NC   (Note that we can uniquely recover inv R (w 2 ) by adding in missing initial segments of oriented dihedral subgroups-in this case, the dihedral generated by {s 1 , s 2 } forces (1, 2) to be added back to the inversion set).
Using the second isomorphism in Proposition 4.48, we conjugate these inversions by w −1 1 to pass to the corresponding standard parabolic subgroup W desR(w1) , obtaining the inversion set {(34), (24)}. This inversion set corresponds to an element with reduced word s 3 s 2 . Therefore, (s 1 s 2 s 3 s 2 s 1 s 2 s 3 ) is mapped to the Garside factorization s 1 s 2 s 3 s 2 · s 3 s 2 .
We note that any 0-forced skips that could be attributed to the w (2) · · · w (m) piece of C c (w) are already 0-forced skips from the w (1) piece of the product, since the support of w (2) · · · w (m) is contained in the support of w (1) . By Proposition 4.17, these 0-forced skips are unaffected by the addition of the piece w (2) · · · w (m) , and therefore account for the first piece of the delta sequence, cu −1 1 by the m = 1 bijection between sortable elements and delta sequences.

m-eralized cluster complexes
In this section, we study an m-eralization of the c-cluster complex. We begin in Section 5.1 with the definition of c-cluster complexes, as introduced by N. Reading in [Rea07b], and its description in terms of subword complexes. We then recall S. Fomin and N. Reading's m-eralization of cluster complexes from [FR05], and generalize their definition to general Coxeter elements in Definitions 5.8 and 5.13. In Section 5.4, we show existence of our generalized cluster complexes by explicitly constructing them as subword complexes. We prove that they are vertexdecomposable and that they are a wedge of spheres. After defining Cambrian rotation and the Cambrian recurrence, in Section 5.6 we give bijections to noncrossing partitions and sortable elements. We conclude by constructing the m-eralized Cambrian lattices on our subword complexes. 5.1. Cluster complexes. Let (W, S) be a finite Coxeter system with root system Φ, let c be a Coxeter element, and let Φ ≥−1 = Φ + ∪ −∆ be the set of almost positive roots. Following N. Reading, the c-compatibility relations is the unique family of relations c on Φ ≥−1 characterized by the following two properties [RS11, Section 5]: (i) for α ∈ ∆ and β ∈ Φ ≥−1 , (ii) for β 1 , β 2 ∈ Φ ≥−1 and s initial in c, where τ s is defined as The c-cluster complex Asso(W, c) is the simplicial complex given by all collections of pairwise c-compatible almost positive roots; in other words, it is the clique complex of the graph on Φ ≥−1 whose edges are given by c . A c-cluster is a facet of the c-cluster complex Asso(W, c)-that is, a maximal subset of almost positive roots which are pairwise c-compatible.
For a fixed Coxeter group W and any two Coxeter elements c and c ′ , the c-and c ′cluster complexes are isomorphic [Rea07a, Proposition 7.2], [BMR + 06, Proposition 4.10]. In crystallographic type, the c-cluster complexes are isomorphic to the cluster complex defined in [FZ02]. 5.2. Cluster complexes as subword complexes. We may encode clusters using E. Tzanaki's explicit decreasing factorizations of Coxeter elements [Tza08, Theorem 1.1], see also [IS10]. In our notation, this encoding has the following form.
The isomorphism Lr c between (positions of ) letters in cw • (c) and almost positive roots is given by sending the letter s i of c = s 1 · · · s i to the simple negative root −α si , and the letter w i of w • (c) to the i-th root in inv(w • (c)).
Remark 5.3. By Proposition 2.6, these two theorems are equivalent. It follows from [KM04, Theorem 2.5] that the c-cluster complex is vertex-decomposable.
This alternative description of the c-cluster complex provides a simple way to prove that all c-cluster complexes are isomorphic. We say that a sequence s 1 , . . . , s p of simple reflections in a sequence of initial letters for a Coxeter element c if s i is initial in c si−1···s1 .
Proof. It is well-known that for any two Coxeter elements c, c ′ , there is a sequence of initial letters s 1 , . . . , s p such that c ′ = c s1···sp . The statement then follows from Proposition 2.14 and Lemma 2.18(1) and (6).
with FR(β) = τ SL τ SR = s∈SL τ s s∈SR τ s , and S = S L ⊔ S R is the bipartition of the simple reflections given in Section 2.1.
The m-eralized compatibility relation is the unique relation on Φ (m) ≥−1 characterized by the following two properties [FR05, Theorem 3.4]: The existence of a binary relation on Φ It is unnecessary to treat bipartite c as a special case. We present an approach to the Fomin-Reading map that puts all Coxeter elements on equal footing, and which we then use to define Asso(W, c) for any Coxeter element c. For s ∈ S, we define the m-eralization of the map τ s from Section 5.1 to be the bijection where we consider colors modulo (m + 1). It follows from the definition that the order of τ (m) s is given by lcm m + 1, 2 , so that it is an involution only for m = 1.
The m-eralized c-compatibility relation c on m-colored almost positive roots is the unique family of relations c on Φ (m) ≥−1 characterized by the following two properties, analogous to the defining properties of the c-compatibility relation given in Section 5.1: ≥−1 and s initial in c, The existence of an m-eralized c-compatibility relation satisfying the above properties is again not immediate. To prove existence, one would need to show that the two properties above uniquely determine the binary relation c and that c is symmetric. We remark that the proofs given in [Rea07a] combined with the ideas in [FR05, Section 6] could probably be adapted to provide a proof of the existence. We prefer the viewpoint in Section 5.4, which gives a construction that directly implies the existence of c .
We begin by showing that a binary relation on Φ (m) ≥−1 satisfying these two properties is unique.
Lemma 5.5. Let c be a Coxeter element and let β ∈ Φ + be a positive root. Then there exists a sequence s 1 , . . . , s p of initial letters in c such that Proof. Since τ Proposition 5.6. Let c be a Coxeter element and let β ∈ Φ + be a positive root. Then there exists a sequence s 1 , . . . , s p of initial letters in c such that τ (m) sp • · · · • τ (m) s1 (β (k) ) = α (m) for a simple root α = α s . In particular, the m-eralized c-compatibility relation is uniquely determined by its two defining properties.
Proof. After moving the root β (k) to a simple root α (k) by the sequence provided by Lemma 5.5, we apply τ (m) sα to obtain α (k−1) . If k = 0, we have sent β (k) to α (m) , and are finished. Otherwise, we again apply such a sequence provided by the lemma to get an initial simple (α ′ ) (k−2) . Repeating k + 1 times therefore yields the desired sequence.
Remark 5.7. We note that, in the proof of Proposition 5.6, we cannot simply apply τ (m) sα as many times as needed after using Lemma 5.5 once. The simple reflection s α becomes final after applying τ (m) sα , and we therefore cannot apply τ (m) sα again immediately after.
Having proven uniqueness of the m-eralized c-compatibility relation, we postpone the proof of existence to instead define the m-eralized c-cluster complex. The following definition assumes the existence of c , which will be established in Section 5.4. Proof. This follows from the observation that, after identifying Φ  Proof. It is a straightforward check that this composition satisfies the two cases in the definition of the Fomin-Reading map FR (m) .
Corollary 5.11. Let c = c L c R be a bipartite Coxeter element. Then Proof. It follows from [FR05, Lemma 6.2] that the m-eralized compatibility relation is invariant under the application of −w • , i.e., and is therefore the m-eralized compatibility relation induced by τ sN in the previous proposition. We conclude that m-eralized c-compatibility relation c for c = c L c R coincides with the m-eralized compatibility relation .
Remark 5.12. As the situation for general m is similar, we give a more structured viewpoint on the relation given in Proposition 5.10 for m = 1. The operation τ s corresponds to the rotation operation described in Proposition 2.15 on the subword complex of Theorem 5.2. In this subword complex, the rotation of the initial n letters induced an automorphism on Asso(W, c) as does the inverse rotation of the final N = ℓ(w • ) letters, and these two automorphisms are obtained from each other by twisting with w • .
The equality of the two operations in Proposition 5.10 should be thought of an m-eralization of this observation. The automorphism on Asso (m) (W ) induced by FR (m) is, by Proposition 5.10, given by τ   c . The three definitions of the m-eralized c-cluster complex are clearly equivalent. On the other hand, we have already given a definition of Asso (m) (W, c) in Definition 5.8, though that definition is yet to be proven to be valid. We thus aim to show that the definition we give here indeed matches the definition given in Definition 5.8.
Remark 5.14. We conjecture that, in the case of this definition, the injection mentioned in Remark 2.13 of the Artin subword complex into corresponding the Coxeter subword complex is actually an isomorphism. That is, we conjecture that     Example 5.16. Parallel to Examples 4.7 and 3.14, we consider Asso     We next show that the m-eralized c-cluster complex is flag.
Proposition 5.18. The m-eralized c-cluster complex in Definition 5.13 is the clique complex of its edges.
Proof. We show that all of the minimal nonfaces of Asso  ∇ (W, c) has r k · · · r 1 ≤ R c, where r i is the ith reflection in inv R (cw m • (c)). It follows from [ABMW06, Lemma 2.1(iv)] and the lattice property of NC(W, c) that for a reduced R-word r 1 · · · r k , r k · · · r 1 ≤ R c ⇔ r i r j ≤ R c for all i > j.
Thus, there are indices k ≥ a > b ≥ 1 such that r a r b ≤ R c, implying that {i a , i b } is also a nonface of Asso (m) ∇ (W, c). Remark 5.19. This statement was inadvertently omitted in [CLS14], where it was only proven that Asso  (W, c). It was implicit in [CLS14] that equality follows, since Asso (1) (W, c) was previously proven to be a simplicial sphere of the same dimension as Asso ≥−1 . We first m-eralize [CLS14, Lemma 5.4], which we then use to show that the m-eralized c-cluster complex is vertex-decomposable. To this end, let c = s 1 · · · s n , and letĉ = s 1 · · ·ŝ k · · · s n be obtained from c by removing the letter s k . Define an order preserving embedding inv R (ĉw m • (ĉ)) ֒→ inv R (cw m • (c)) by fixing the first k − 1 reflections of inv R (ĉw m • (ĉ)) (corresponding to the initial segment s 1 · · · s k−1 of c) and conjugating the remaining reflections by s 1 · · · s k .
As we have seen in Example 2.16, we cannot use the usual theory of subword complexes to conclude that the generalized subword complex of Definition 5.13 is vertex-decomposable. Nevertheless, the m-eralized cluster complex Asso (m) (W ) is vertex-decomposable, as shown by C. Athanasiadis and E. Tzanaki in [AT08]. Using the previous lemma, we give a short proof in Proposition 5.25. We emphasize that we will later need vertex-decomposability to prove Theorem 5.20, and we therefore may not simply deduce this property from previous work.
Proposition 5.25. Let c = s 1 · · · s n , let Q = s 1 · · · s p be an initial segment of the word w m • (c), and let w = s 1 · · · s p ∈ W be the corresponding element in W . Then Sub S (cQ, w, p) is vertex-decomposable. In particular, Asso (m) (W, c) is vertexdecomposable.
Proof. We show that both the link and the deletion of the first vertex are vertexdecomposable subword complexes by simultaneous induction on the rank of W and on the length of Q.
The facets of the link of the first vertex in Sub S (cQ, w, p) are given by I \ 1 : I a facet of Sub S (cQ, w, p) with 1 ∈ I .
Whenĉ = s 2 · · · s n , Sub R (ĉQ,ĉ −1 ) ∼ = Sub R (ĉQ,ĉ −1 ) forQ as given in Lemma 5.24. This complex is vertex-decomposable by induction on the rank of W . We now consider the deletion. Let I be a facet of Sub S (cQ, w, p) with 1 ∈ I, so that I \ 1 is a face of the deletion of this letter. To set up an inductive argument on the length of Q, we show that there is a facet J of Sub S (cQ, w, p) with J \ j = I \ 1 and j > 1. Letting c ′ = s −1 1 cs 1 , the facet J is naturally a facet of Sub S (c ′ s 2 · · · s p , w, p). We first extend Q to w m • (c) and observe that every facet of Sub S (cQ, w, p) is also a facet of Asso (m) (W, c). Treating I as a facet of Asso (m) (W, c), we may then flip 1 ∈ I to the position j for which r I (j) = α (0) s1 .
After performing this flip, if there is a position k ∈ I with j < k, then this flip did not cause the facet to leave the initial segment cQ. Otherwise, we claim that the facet I was equal to the first n letters of cQ, and that the position j was the position of the first letter of Q. By assumption, no position of a letter to the right of j is inside I. Since the product of the root configuration corresponding to the letters of I is c −1 , we can use Lemma 2.18(5) to read the letters of w m • (c) from right to left until we pass j. From the form of w m • (c), it is clear that the right-most possible location of the colored positive root α (0) s1 is the left-most position of Q. Therefore, I is indeed the initial copy of c. We can now replace c by s −1 1 cs 1 , and deduce the statement by induction on the length of Q.
Finally, we note that the two base cases of W of rank 1 and the empty word Q are trivially vertex-decomposable.
Corollary 5.26. The lexicographic order of the facets of Asso (m) (W, c) is a shelling order.
Proof. The order on the vertices in the vertex-decomposition in Proposition 5.25 is the usual order. The lexicographic order on Asso (m) (W, c) is therefore a shelling order.
We now prove Proposition 5.22.
Proof of Proposition 5.22. Using the root configuration as defined in Equation (2) together with the fact that Asso (m) (W, c) is vertex-decomposable-and therefore shellable-we obtain that ≥−1 , using the same proof as [CLS14, Proposition 5.3]. We require shellability to ensure that all facets of Asso (m) (W, c) containing the first letter are connected by flips not using this first letter.
Proof of Theorem 5.20. Proposition 5.6 showed that the two defining properties uniquely determine c . Propositions 5.21 and 5.22 show that ′ c satisfies the defining properties of an m-eralized c-compatibility relation, so that ′ c and c coincide. We conclude this section by defining the support of a facet of the m-eralized c-cluster complex.
Proof. The statement follows from the facts that w • is an involution on S, N = nh/2 and the length of the word cw m • (c) is n + mN . We modify the shift operator Shift s to produce the m-eralized c-Cambrian recurrence on facets of Asso    Generalizing [Rea07a,Theorem 8.1], we conclude this section with a direct bijection between Sort (m) (W, c) and Asso (m) (W, c).
For w ∈ Sort (m) (W, c), consider the c-sorting w(c) = s 1 · · · s p . For s ∈ supp(w), let β (ℓs) s be the colored positive root s 1 · · · s k−1 (α (0) s k ), where s k is the last occurrence of the letter s in w(c).  of m-colored positive roots is a bijection that respects Cambrian rotation, the Cambrian recurrence, and support.
We prove this theorem by showing that both satisfy the Cambrian recurrence. The notational similarity to the skip set is chosen to emphasize its very similar recurrence. For simplicity, denote by ν          By symmetry and by applying Shift a suitable number of times, we may therefore assume that the given flip is positive and uses the initial letter s (0) in the search word for Asso (m) (W, c), where s is initial in c.
We now choose to work in Asso     Remark 5.45. For any two Coxeter elements c and c ′ , the underlying unoriented flip graphs of the complexes Asso (m) (W, c) and Asso (m) (W, c ′ ) are isomorphic. The isomorphism is induced by the shift operation, and is given by the isomorphism in Proposition 5.17. In particular, the map Camb c is a graph automorphism. For m = 1, the increasing flip graph coincides with the Hasse diagram of the Cambrian poset, while for m ≥ 2 the increasing flip graph is no longer transitively reduced. Therefore, the shift operation does not induce a isomorphism between the unoriented Hasse diagrams of Camb