Subword complexes, cluster complexes, and generalized multiassociahedra
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Abstract
In this paper, we use subword complexes to provide a uniform approach to finitetype cluster complexes and multiassociahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multicluster complex. For k=1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multicluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multitriangulations and centrally symmetric multitriangulations, respectively. Furthermore, we show that the multicluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multicluster complex.
Keywords
Subword complex Cluster complex Generalized associahedron Multitriangulation Auslander–Reiten quiver Coxeter–Catalan combinatorics1 Introduction
Cluster complexes were introduced by S. Fomin and A. Zelevinsky to encode exchange graphs of cluster algebras [20]. N. Reading then showed that the definition of cluster complexes can be extended to all finite Coxeter groups [48, 49]. In this article, we present a new combinatorial description of cluster complexes using subword complexes. These were introduced by A. Knutson and E. Miller, first in type A to study the combinatorics of determinantal ideals and Schubert polynomials [35], and then for all Coxeter groups in [34]. We provide, for any finite Coxeter group W and any Coxeter element c∈W, a subword complex which is isomorphic to the ccluster complex of the corresponding type, and we thus obtain an explicit typefree characterization of cclusters. This characterization generalizes a description for crystallographic types obtained by K. Igusa and R. Schiffler in the context of cluster categories [29]. The present approach allows us to define a new family of simplicial complexes by introducing an additional parameter k, such that one obtains ccluster complexes for k=1. In type A, this simplicial complex turns out to be isomorphic to the simplicial complex of multitriangulations of a convex polygon which was described by C. Stump in [62] (see also [60]), and, in a similar manner, by V. Pilaud and M. Pocchiola in the framework of sorting networks [43]. In type B, we obtain that this simplicial complex is isomorphic to the simplicial complex of centrally symmetric multitriangulations of a regular convex polygon. Therefore, we call them multicluster complexes. Besides, there is a “naive” way of generalizing the cluster complex by considering the simplicial complex of sets of almost positive roots that do not contain any k+1 pairwise not compatible roots. In fact, this alternative definition lacks basic properties of the cluster complex, see Remark 2.9. The multicluster complexes introduced in this paper are different from generalized cluster complexes as defined by S. Fomin and N. Reading [18]. In the generalized cluster complex, the vertices are given by the simple negative roots together with several distinguished copies of the positive roots, while the vertices of the multicluster complex correspond to the positive roots together with several distinguished copies of the simple negative roots. Multicluster complexes turn out to be intimately related to Auslander–Reiten quivers and repetition quivers [24]. In particular, the Auslander–Reiten translate on facets of multicluster complexes in types A and B corresponds to cyclic rotation of (centrally symmetric) multitriangulations. Furthermore, multicluster complexes uniformize questions about multitriangulations, subword complexes, and cluster complexes. One important example concerns the open problem of realizing the simplicial complexes of (centrally symmetric) multitriangulations and spherical subword complexes as boundary complexes of convex polytopes.
In Sect. 2, we recall the various objects in question, namely multitriangulations, subword complexes, and cluster complexes. Moreover, the main results are presented, and the multicluster complex is defined (Definition 2.5). In Sect. 3, we study flips on spherical subword complexes and present two natural isomorphisms between subword complexes whose words differ by commutation or by rotation of letters. In Sect. 4, we prove that the multicluster complex is independent of the choice of the Coxeter element (Theorem 2.6). Section 5 contains a proof that the multicluster complex is isomorphic to the cluster complex for k=1 (Theorem 2.2). In Sect. 6, we discuss possible generalizations of associahedra using subword complexes; we review known results about polytopal realizations, prove polytopality of multicluster cluster complexes of rank 2 (Theorem 6.1), and prove that the multicluster complex is universal in the sense that every spherical subword complex is the link of a face of a multicluster complex (Theorem 2.15). Section 7 contains a combinatorial description of the sorting words of the longest element of finite Coxeter groups (Theorem 7.2) and an alternative definition of multicluster complexes in terms of the strong intervening neighbors property (Theorem 2.7). In Sect. 8, we connect multicluster complexes to Auslander–Reiten quivers and repetition quivers, and use this connection to introduce an action on vertices and facets of multicluster complexes generalizing the natural rotation action on multitriangulations in type A. Finally, in Sect. 9, we discuss open problems and questions arising in the context of multicluster complexes.
In a subsequent paper, C. Stump and V. Pilaud study the geometry of subword complexes and use the theory developed in the present paper to describe the connections to Coxetersortable elements, and how to recover Cambrian fans, Cambrian lattices, and the generalized associahedra purely in terms of subword complexes [45].
2 Definitions and main results
2.1 Multitriangulations
Let Δ _{ m } be the simplicial complex with vertices being diagonals of a convex mgon and faces being subsets of noncrossing diagonals. Its facets correspond to triangulations (i.e., maximal subsets of diagonals which are mutually noncrossing). This simplicial complex is the boundary complex of the dual associahedron [13, 22, 25, 26, 37, 38], we refer to the recent book [39] for a detailed treatment of the history of associahedra. The complex Δ _{ m } can be generalized using a positive integer k with 2k+1≤m: define a (k+1)crossing to be a set of k+1 diagonals which are pairwise crossing. A diagonal is called krelevant if it is contained in some (k+1)crossing, that is, if there are at least k vertices of the mgon on each side of the diagonal. The complex Δ _{ m,k } is the simplicial complex of (k+1)crossing free sets of krelevant diagonals. Its facets are given by ktriangulations (i.e., maximal subsets of diagonals which do not contain a (k+1)crossing), without considering kirrelevant diagonals. The reason for restricting the set of diagonals is that including all diagonals that are not krelevant would yield the join of Δ _{ m,k } and an mksimplex. This simplicial complex has been studied by several authors, see, e.g., [14, 31, 32, 36, 41, 55, 62]; an interesting recent treatment of ktriangulations using complexes of star polygons can be found in [44]. We refer to this simplicial complex as the simplicial complex of multitriangulations.
Example 2.1
For m=5 and k=1, we get Q=(q _{1},q _{2},q _{3},q _{4},q _{5})=(s _{2},s _{1},s _{2},s _{1},s _{2}). By cyclically labeling the vertices of the pentagon with the integers {0,…,4}, the bijection sends the (position of the) letter q _{ i } to the ith entry of the list of ordered diagonals [0,2],[0,3],[1,3],[1,4],[2,4]. On one hand, two cyclically consecutive diagonals in the list form a triangulation of the pentagon. On the other hand, the complement of two cyclically consecutive letters of Q form a reduced expression for \([3,2,1] = s_{1} s_{2} s_{1} = s_{2} s_{1} s_{2} \in \mathcal{S}_{3}\).
The main objective of this paper is to describe and study a natural generalization of multitriangulations to finite Coxeter groups.
2.2 Subword complexes
2.3 Cluster complexes
 (i)for s∈S and β∈Φ _{≥−1},$$\alpha_s \parallel_c \beta\quad \Longleftrightarrow \quad \beta\in(\varPhi _{\langle s \rangle})_{\geq1}; $$
 (ii)for β _{1},β _{2}∈Φ _{≥−1} and s being initial in c,$$\beta_1 \parallel_c \beta_2 \quad \Longleftrightarrow \quad \sigma_s(\beta_1) \parallel_{\mathit{scs}} \sigma_s(\beta_2). $$
2.4 Main results
We are now in the position to state the main results of this paper and to define the central object, the multicluster complex. Let c=(c _{1},…,c _{ n }) be a reduced expression for a Coxeter element c∈W, and let w _{∘}(c)=(w _{1},…,w _{ N }) be the lexicographically first subword of c ^{∞} that represents a reduced expression for the longest element w _{∘}∈W. The word w _{∘}(c) is called csorting word for w _{∘}, see [48]. The first theorem (proved in Sect. 5) gives a description of the cluster complex as a subword complex.
Theorem 2.2
The subword complex Δ(cw _{∘}(c),w _{∘}) is isomorphic to the ccluster complex. The isomorphism is given by sending the letter c _{ i } of c to the negative root \(\alpha_{c_{i}}\) and the letter w _{ i } of w _{∘}(c) to the positive root \(w_{1} \cdots w_{i1}(\alpha_{w_{i}})\).
As an equivalent statement, we obtain the following explicit description of the ccompatibility relation.
Corollary 2.3
A subset C of Φ _{≥−1} is a ccluster if and only if the complement of the corresponding subword in cw _{∘}(c)=(c _{1},…,c _{ n },w _{1},…,w _{ N }) represents a reduced expression for w _{∘}.
This description was obtained independently by K. Igusa and R. Schiffler [29] for finite crystallographic root systems in the context of cluster categories [29, Theorem 2.5]. They use results of W. CrawleyBeovey and C.M. Ringel saying that the braid group acts transitively on isomorphism classes of exceptional sequences of modules over a hereditary algebra, see [29, Sect. 2]. K. Igusa and R. Schiffler then show combinatorially that the braid group acting on sequences of elements in any Coxeter group W of rank n acts as well transitively on all sequences of n reflections whose product is a given Coxeter element [29, Theorem 1.4]. They then deduce Corollary 2.3 in crystallographic types from these two results, see [29, Theorem 2.5]. The present approach holds uniformly for all finite Coxeter groups and is developed purely in the context of Coxeter group theory. We study the connections to the work of K. Igusa and R. Schiffler more closely in Sect. 8. In the particular case of bipartite Coxeter elements, as defined in Sect. 8.3 below, a similar description as in Corollary 2.3 was as well obtained by T. Brady and C. Watt in [11] in the context of the geometry of noncrossing partitions.^{1}
Example 2.4
Inspired by results in [62] and [43], we generalize the subword complex in Theorem 2.2 by considering the concatenation of k copies of the word c. In type A, this generalization coincides with the description of the complex Δ _{ m,k } given in [43] in a different language.
Definition 2.5
The multicluster complex \(\varDelta ^{k}_{c}(W)\) is the subword complex Δ(c ^{ k } w _{∘}(c),w _{∘}).
Multicluster complexes are in fact independent of the Coxeter element c. In particular, we reobtain that all ccluster complexes are isomorphic (see Sect. 4 for the proof).
Theorem 2.6
All multicluster complexes \(\varDelta ^{k}_{c}(W)\) for the various Coxeter elements are isomorphic.
The following two results give alternative descriptions of multicluster complexes. A word Q=(q _{1},…,q _{ r }) in S has the intervening neighbors property, if all noncommuting pairs s,t∈S alternate within Q, see [15, Sect. 3] and [59, Proposition 2.1]. Let ψ:S→S be the involution given by \(\psi(s)=w_{\circ}^{1} s w_{\circ}\), and extend ψ to words as ψ(Q)=(ψ(q _{1}),…,ψ(q _{ r })). We say that Q has the strong intervening neighbors property (SINproperty) if Qψ(Q)=(q _{1},…,q _{ r },ψ(q _{1}),…,ψ(q _{ r })) has the intervening neighbors property and if in addition the Demazure product δ(Q) is w _{∘}. Two words coincide up to commutations if they can be obtained from each other by a sequence of interchanges of consecutive commuting letters. The next theorem (proved in Sect. 7) characterizes all words that are equal to c ^{ k } w _{∘}(c) up to commutations.
Theorem 2.7
A word in S has the SINproperty if and only if it is equal to c ^{ k } w _{∘}(c), up to commutations, for some Coxeter element c and some nonnegative integer k.
Proposition 2.8
Proof
The proof follows the lines of the proof of [29, Lemma 3.2]. A direct calculation shows that \(t_{\ell_{1}}\cdots t_{\ell_{kn}} q_{1}q_{2}\cdots q_{kn+N}\) equals the product of all letters in cw _{∘}(c) not in \(\{q_{\ell_{1}},\ldots ,q_{\ell_{kn}}\}\). We get that \(\{q_{\ell_{1}},\ldots,q_{\ell_{kn}}\} \) is a facet of \(\varDelta ^{k}_{c}(W)\) if and only if \(t_{\ell_{1}}\cdots t_{\ell _{kn}} q_{1}q_{2}\cdots q_{kn+N} = w_{\circ}\). As q _{1} q _{2}⋯q _{ kn+N }=c ^{ k } w _{∘}, the statement follows. □
Remark 2.9
Note that there is as well a “naive” way of extending the notion of cluster complexes. Consider the simplicial complex on the set of almost positive roots whose faces are given by the sets that do not contain any subset of k+1 pairwise not compatible roots. In type A, this complex gives rise to the simplicial complex of multitriangulations of a convex polygon as desired. However, this simplicial complex lacks basic properties of cluster complexes in general; in type B _{3}, it is not pure. In this case, the maximal faces have cardinality 6 or 7. A similar phenomenon was observed in [43, Remark 29], where the authors suggest that subword complexes of type A (viewed as pseudoline arrangements) are the right objects to define “multipseudotriangulations” and explain that the approach using pairwise crossings does not work.
The correspondence between the concepts of diagonals, multitriangulations and flips of multitriangulations in Δ _{ m,k }, and the multicluster complex \(\varDelta ^{k}_{c}(A_{m2k1})\)
Δ _{ m,k }  \(\varDelta ^{k}_{c}(A_{m2k1})\)  

Vertices  kRelevant diagonals of a convex mgon  Letters of Q=c ^{ k } w _{∘}(c) 
Facets  Maximal sets of krelevant diagonals without (k+1)crossings  P⊂Q such that Q∖P is a reduced expression for w _{∘} 
Simplices  Sets of krelevant diagonals without (k+1)crossings  P⊂Q such that Q∖P contains a reduced expression for w _{∘} 
Ridges  Flips between two ktriangulations  Facet flips using Lemma 3.3 
Also in type B, we obtain a previously known object, namely the simplicial complex \(\varDelta _{m,k}^{\mathrm{sym}}\) of centrally symmetric ktriangulations of a regular convex 2mgon. The vertices of this complex are pairs of centrally symmetric krelevant diagonals, and a collection of vertices form a face if and only if the corresponding diagonals do not contain a (k+1)crossing. This simplicial complex was studied in algebraic and combinatorial contexts in [51, 63]. We refer to Sect. 6.3 for a proof of Theorem 2.10.
Theorem 2.10
The multicluster complex \(\varDelta ^{k}_{c}(B_{mk})\) is isomorphic to the simplicial complex of centrally symmetric ktriangulations of a regular convex 2mgon.
The description of the simplicial complex of centrally symmetric multitriangulations as a subword complex provides straightforward proofs of nontrivial results about centrally symmetric multitriangulations.
Corollary 2.11
 (i)
All centrally symmetric ktriangulations of a regular convex 2mgon contain exactly mk relevant (centrally) symmetric pairs of diagonals, of which k are diameters.
 (ii)
For any centrally symmetric ktriangulation T and any krelevant symmetric pair of diagonals d∈T, there exists a unique krelevant symmetric pair of diagonals d′ not in T such that T′=(T∖{d})∪{d′} is again a centrally symmetric ktriangulation. The operation of interchanging a symmetric pair of diagonals between T and T′ is called symmetric flip.
 (iii)
All centrally symmetric ktriangulations of a 2mgon are connected by symmetric flips.
The generalization of the concept of diagonals, multitriangulations and flips of multitriangulations to the Coxeter group of type B _{ n }
\(\varDelta _{m,k}^{\mathrm{sym}}\)  \(\varDelta ^{k}_{c}(B_{mk})\)  

Vertices  kRelevant symmetric pairs of diagonals of a regular convex 2mgon  Letters of Q=c ^{ k } w _{∘}(c)=c ^{ m } 
Facets  Maximal sets of krelevant centrally symmetric diagonals without (k+1)crossings  P⊂Q such that Q∖P is a reduced expression for w _{∘} 
Simplices  Sets of krelevant symmetric pairs of diagonals without (k+1)crossings  P⊂Q such that Q∖P contains a reduced expression for w _{∘} 
Ridges  Symmetric flips between two centrally symmetric diagonals without (k+1)crossings  Facet flips using Lemma 3.3 
Example 2.12
Using algebraic techniques, D. Soll and V. Welker proved that \(\varDelta _{m,k}^{\mathrm{sym}}\) is a (mod 2)homologysphere [63, Theorem 10]. The subword complex description in Theorem 2.10 and the results by A. Knutson and E. Miller [34, Theorem 2.5 and Corollary 3.8] imply the following stronger result.
Corollary 2.13
The simplicial complex of centrally symmetric ktriangulations of a regular convex 2mgon is a vertexdecomposable simplicial sphere.
This result, together with the proof of [63, Conjecture 13] given in [51],^{2} implies the following conjecture by D. Soll and V. Welker.
Corollary 2.14
[63, Conjecture 17]
For the termorder ⪯ defined in [63, Sect. 7], the initial ideal \(\operatorname{in}_{\preceq}(I_{n,k})\) of the determinantal ideal I _{ n,k } defined in [63, Sect. 3] is spherical.
We finish this section by describing all spherical subword complexes in terms of faces of multicluster complexes (see Sect. 6.5 for the proofs). This generalizes the universality of the multiassociahedron presented in [46, Proposition 5.6] to finite Coxeter groups.
Theorem 2.15
A simplicial sphere can be realized as a subword complex of a given finite type W if and only if it is the link of a face of a multicluster complex \(\varDelta ^{k}_{c}(W)\).
The previous theorem can be obtained for any family of subword complexes for which arbitrary large powers of c appear as subwords. However, computations seem to indicate that the multicluster complex maximizes the number of facets among subword complexes Δ(Q,w _{∘}) with word Q of the same size. We conjecture that this is true in general, see Conjecture 9.8. We also obtain the following corollary.
Corollary 2.16
 (i)
Every spherical subword complex of type W is polytopal.
 (ii)
Every multicluster complex of type W is polytopal.
3 General results on spherical subword complexes
Before proving the main results, we discuss several properties of spherical subword complexes in general which are not specific to multicluster complexes. Throughout this section, we let Q=(q _{1},…,q _{ r }) be a word in S and π=δ(Q).
3.1 Flips in spherical subword complexes
Lemma 3.1
(Knutson–Miller)
Let F be a facet of Δ(Q,δ(Q)). For any vertex q∈F, there exists a unique vertex q′∈Q∖F such that (F∖{q})∪{q′} is again a facet.
Proof
This follows from the fact that Δ(Q,δ(Q)) is a simplicial sphere [34, Corollary 3.8]. See [34, Lemma 3.5] for an analogous reformulation. □
Such a move between two adjacent facets is called flip. Next, we describe how to find the unique vertex q′∉F corresponding to q∈F. For this, we introduce the notion of root functions.
Definition 3.2
The root function r _{ F }:Q→Φ associated to a facet F of Δ(Q,π) sends a letter q∈Q to the root r _{ F }(q):=w _{ q }(α _{ q })∈Φ, where w _{ q }∈W is given by the product of the letters in the prefix of \(Q\setminus F=(q_{i_{1}}, \dots, q_{i_{\ell}})\) that appears on the left of q in Q, and where α _{ q } is the simple root associated to q.
Lemma 3.3
Let F, q, and q′ be as in Lemma 3.1. The vertex q′ is the unique vertex not in F for which r _{ F }(q′)∈{±r _{ F }(q)}.
Proof
Since \(q_{i_{1}}\dots q_{i_{\ell}}\) is a reduced expression for π=δ(Q), the set \(\{\mathsf{r}_{F}({q_{i_{1}}}), \dots, \mathsf{r}_{F}({q_{i_{\ell}}})\}\) is equal to the inversion set \(\operatorname{inv}(\pi) = \{ \alpha_{i_{1}}, q_{i_{1}}(\alpha_{i_{2}}), \ldots, q_{i_{1}}\cdots q_{i_{\ell1}}(\alpha _{i_{\ell}}) \}\) of π, which only depends on π and not on the chosen reduced expression. In particular, any two elements in this set are distinct. Notice that the root r _{ F }(q) for q∈F is, up to sign, also contained in \(\operatorname{inv}(\pi)\); otherwise it would contradict the fact that the Demazure product of Q is π. If we insert q into the reduced expression of π, the exchange property in Coxeter groups implies that we have to delete the unique letter q′ that corresponds to the same root, with a positive sign if it appears on the right of q in Q, or with a negative sign otherwise. The resulting word is again a reduced expression for π. □
Remark 3.4
In the case of cluster complexes, this description can be found in [29, Lemma 2.7].
Example 3.5
The following lemma describes the relation between the root functions of two facets connected by a flip.
Lemma 3.6
Proof

If p is on the left of q or p=q, then \(w_{p} =w_{p} '\) and r _{ F }(p)=r _{ F′}(p).

If p is between q and q′ or p=q′, then \(w_{p} '\) can be obtained from w _{ p } by adding the letter q at its corresponding position. This addition is the result of multiplying w _{ p } by \(t_{q}=w_{q}qw_{q}^{1}\) on the left, i.e., \(w_{p} '=t_{q}w_{p} \). Therefore, r _{ F }(p)=t _{ q }(r _{ F′}(p)).

If p is on the right of q′, consider the reflection \(t_{q'}=w_{q'}q'w_{q'}^{1}\) where w _{ q′} is the product of the letters in the prefix of Q∖F that appears on the left of q′. By the same argument, one obtains that \(w_{p} '=t_{q}t_{q'}w_{p} \). In addition, t _{ q }=t _{ q′} because they correspond to the unique reflection orthogonal to the roots r _{ F }(q) and r _{ F }(q′), which are up to sign equal by Lemma 3.3. Therefore, \(w_{p} '=w_{p} \) and r _{ F′}(p)=r _{ F }(p).
3.2 Isomorphic spherical subword complexes
We now reduce the study of spherical subword complexes in general to the case where δ(Q)=π=w _{∘} and give two operations on the word Q giving isomorphic subword complexes.
Theorem 3.7
Every spherical subword complex Δ(Q,π) is isomorphic to Δ(Q′,w _{∘}) for some word Q′ such that δ(Q′)=w _{∘}.
Proof
Let r be a reduced word for π ^{−1} w _{∘}=δ(Q)^{−1} w _{∘}∈W. Moreover, define the word Q′ as the concatenation of Q and r. By construction, the Demazure product of Q′ is w _{∘}, and every reduced expression of w _{∘} in Q′ must contain all the letters in r. The reduced expressions of w _{∘} in Q′ are given by reduced expressions of π in Q together with all the letters in r. Therefore, the subword complexes Δ(Q,π) and Δ(Q′,w _{∘}) are isomorphic. □
Recall the involution ψ:S→S given by \(\psi(s)=w_{\circ}^{1} s w_{\circ}\). This involution was used in [8] to characterize isometry classes of the cgeneralized associahedra. Define the rotated word \(Q_{\stackrel {\circlearrowleft }{s}}\) or the rotation of Q=(s,q _{2},…,q _{ r }) along the letter s as (q _{2},…,q _{ r },ψ(s)). The following two propositions are direct consequences of the definition of subword complexes.
Proposition 3.8
If two words Q and Q′ coincide up to commutations, then Δ(Q,π)≅Δ(Q′,π).
Proof
The isomorphism between Δ(Q,π) and Δ(Q′,π) is induced by reordering the letters of Q to obtain Q′. □
Proposition 3.9
Let Q=(s,q _{2},…,q _{ r }). Then \(\varDelta (Q,w_{\circ})\cong \varDelta (Q_{\stackrel {\circlearrowleft }{s}}, w_{\circ})\).
Proof
The isomorphism between Δ(Q,w _{∘}) and \(\varDelta (Q_{\stackrel {\circlearrowleft }{s}},w_{\circ})\) is induced by sending q _{ i } to q _{ i } for 2≤i≤r and the initial s to the final ψ(s). The results follows using the fact that sw _{∘}=w _{∘} ψ(s). □
4 Proof of Theorem 2.6
In this section, we prove that all multicluster complexes for the various Coxeter elements are isomorphic. This result relies on the theory of sorting words and sortable elements introduced by N. Reading in [48]. The csorting word for w∈W is the lexicographically first (as a sequence of positions) subword of c ^{∞}=ccc… which is a reduced word for w. We use the following result of D. Speyer.
Lemma 4.1
[59, Corollary 4.1]
The longest element w _{∘}∈W can be expressed as a reduced prefix of c ^{∞} up to commutations.
The next lemma unifies previously known results; the first statement it trivial, the second statement can be found in [59, Sect. 4], and the third statement is equivalent to [27, Lemma 1.6].
Lemma 4.2
 (i)
(p _{2},…,p _{ r }) is a prefix of (c′)^{∞} up to commutations, where c′ denotes a word for the Coxeter element c′=scs,
 (ii)
if p=sp _{2}⋯p _{ r } is reduced, then \(\bf p\) is the csorting word for p up to commutations,
 (iii)
if sp _{2}⋯p _{ r } s′ is reduced for some s′∈S, then p is a prefix of the csorting word for ps′ up to commutations.
Proposition 4.3
Let s be initial in c, and let w _{∘}(c)=(s,w _{2},…,w _{ N }) be the csorting word of w _{∘} up to commutations. Then, (w _{2},…,w _{ N },ψ(s)) is the scssorting word of w _{∘} up to commutations.
Proof
By Lemma 4.1, the element w _{∘} can be written as a prefix of c ^{∞}. By Lemma 4.2, this prefix is equal to the csorting of w _{∘}, which we denote by w _{∘}(c). Let scs denote the word for the Coxeter element scs. By Lemma 4.2 (i), the word (w _{2},…,w _{ N }) is a prefix of (scs)^{∞}, and by (ii) it is the scssorting word for w _{2}⋯w _{ N }. By the definition of ψ, the word (w _{2},…,w _{ N },ψ(s)) is a reduced expression for w _{∘}. Lemma 4.2 (iii) with the word (w _{2},…,w _{ N }) and ψ(s) implies that (w _{2},…,w _{ N },ψ(s)) is the scssorting word for w _{∘} up to commutations. □
Remark 4.4
In [52], N. Reading and D. Speyer present a uniform approach to the theory of sorting words and sortable elements. This approach uses an antisymmetric bilinear form, which is used to extend many results to infinite Coxeter groups. In particular, the previous proposition can be easily deduced from [52, Lemma 3.8].
We are now in the position to prove that all multicluster complexes for the various Coxeter elements are isomorphic.
Proof of Theorem 2.6
Let c and c′ be two Coxeter elements such that c′=scs for some initial letter s of c, and let c and \(\mathbf{c'}\) denote reduced words for c and c′, respectively. Moreover, let Q _{ c }=c ^{ k } w _{∘}(c), and Q _{ c′}=(c′)^{ k } w _{∘}(c′). By Proposition 3.8, we can assume that Q _{ c }=(s,c _{2},…,c _{ n })^{ k }⋅(s,w _{2},…,w _{ N }), and by Proposition 4.3, we can also assume that Q _{ c′}=(c _{2},…,c _{ n },s)^{ k }⋅(w _{2},…,w _{ N },ψ(s)). Therefore, \(Q_{c'} = (Q_{c})_{\stackrel {\circlearrowleft }{s}}\), and Proposition 3.9 implies that the subword complexes Δ(Q _{ c },w _{∘}) and Δ(Q _{ c′},w _{∘}) are isomorphic. Since any two Coxeter elements can be obtained from each other by conjugation of initial letters (see [23, Theorem 3.1.4]), the result follows. □
5 Proof of Theorem 2.2
5.1 Proof of condition (i)
The following theorem implies that Δ(cw _{∘}(c),w _{∘}) satisfies the initial condition.
Theorem 5.1
{−α _{ s },β} is a face of the subword complex Δ(cw _{∘}(c),w _{∘}) if and only if β∈(Φ _{〈s〉})_{≥−1}.
We prove this theorem in several steps.
Lemma 5.2
 (i)
for every q∈F with q≠c _{ i }, \(r_{F}(q)\in\varPhi_{\langle c_{i} \rangle}\),
 (ii)
for every q∈cw _{∘}(c), \(r_{F}(q)\in\varPhi_{\langle c_{i} \rangle }\) if and only if \(\mathsf{Lr}_{c}(q)\in(\varPhi_{\langle c_{i} \rangle})_{\geq1}\).
Proof
For the second part of the lemma, notice that the set \(\{ q \in { \mathbf{c}{\mathbf {w} _{\circ}(\mathbf{c})}}: r_{F}(q) \in\varPhi_{\langle c_{i} \rangle} \}\) is invariant for every facet F containing c _{ i }. In particular, if F=c, this set is equal to \(\{q \in { \mathbf{c}{\mathbf {w} _{\circ}(\mathbf{c})}}: \mathsf{Lr}_{c}(q) \in(\varPhi_{\langle c_{i} \rangle })_{\geq1} \}\). Therefore, \(r_{F}(q)\in\varPhi_{\langle c_{i} \rangle}\) if and only if \(\mathsf{Lr}_{c}(q) \in(\varPhi_{\langle c_{i} \rangle})_{\geq1}\). □
Proposition 5.3
If a facet F of Δ(cw _{∘}(c),w _{∘}) contains c _{ i } and q≠c _{ i }, then \(\mathsf{Lr}_{c}(q)\in(\varPhi_{\langle c_{i} \rangle})_{\geq1}\).
Proof
This proposition is a direct consequence of Lemma 5.2. □
Next, we consider the parabolic subgroup \(W_{\langle c_{i} \rangle}\) obtained by removing the generator c _{ i } from S.
Lemma 5.4
Let c′ be the Coxeter element of the parabolic subgroup \(W_{\langle c_{i} \rangle}\) obtained from c by removing the generator c _{ i }. Consider the word \(\widehat{Q} = {\mathbf{c}' w_{\circ}}(\mathbf{c})\) obtained by deleting the letter c _{ i } from Q=cw _{∘}(c), and let Q′=c′w _{∘}(c′). Then, the subword complexes \(\varDelta (\widehat{Q},w_{\circ})\) and \(\varDelta (Q',w'_{\circ})\) are isomorphic.
Proof
The next lemma states that every letter in cw _{∘}(c) is indeed a vertex of Δ(cw _{∘}(c),w _{∘}).
Lemma 5.5
Every letter in cw _{∘}(c) is contained in some facet of Δ(cw _{∘}(c),w _{∘}).
Proof
Write the word Q=cw _{∘}(c) as the concatenation of c and the cfactorization of w _{∘}, i.e., \(Q= \mathbf{c}\mathbf{c}_{K_{1}} \mathbf{c}_{K_{2}} \cdots \mathbf{c}_{K_{r}}\), where K _{ i }⊆S for 1≤i≤r and c _{ I }, with I⊆S, is the Coxeter element of W _{ I } obtained from c by keeping only letters in I. Since w _{∘} is csortable, see [48, Corollary 4.4], the sets K _{ i } form a decreasing chain of subsets of S, i.e., K _{ r }⊆K _{ r−1}⊆⋯⊆K _{1}⊆S. This implies that the word \(\mathbf{c}\mathbf{c}_{K_{1}} \cdots \widehat{\mathbf{c}}_{K_{i}} \cdots \mathbf{c}_{K_{r}}\) contains a reduced expression for w _{∘} for any 1≤i≤r. Thus, all letters in \(\mathbf{c}_{K_{i}}\) are indeed vertices. □
Proposition 5.6
For every q∈cw _{∘}(c) satisfying \(\mathsf{Lr}_{c}(q) \in(\varPhi_{\langle c_{i} \rangle})_{\geq1}\), there exists a facet of Δ(cw _{∘}(c),w _{∘}) that contains both c _{ i } and q.
Proof
Consider the parabolic subgroup \(W_{\langle c_{i} \rangle}\) obtained by removing the letter c _{ i } from S, and let \(\widehat{Q}\) and Q′ be the words as defined in Lemma 5.4. Since \(\varDelta (\widehat{Q},w_{\circ})\) and \(\varDelta (Q',w'_{\circ})\) are isomorphic, applying Lemma 5.5 to \(\varDelta (Q',w_{\circ}')\) completes the proof. □
Proof of Theorem 5.1
Taking c _{ i }=s, −α _{ s }=Lr _{ c }(c _{ i }), and β=Lr _{ c }(q), the two directions of the equivalence follow from Propositions 5.3 and 5.6. □
5.2 Proof of condition (ii)
The following theorem proves condition (ii).
Theorem 5.7
Let β _{1},β _{2}∈Φ _{≥−1}, and let s be an initial letter of a Coxeter element c. Then, {β _{1},β _{2}} is a face of the subword complex Δ(cw _{∘}(c),w _{∘}) if and only if {σ _{ s }(β _{1}),σ _{ s }(β _{2})} is a face of the subword complex Δ(c′w _{∘}(c′),w _{∘}) with c′=scs.
Proof
6 Generalized multiassociahedra and polytopality of spherical subword complexes
Dictionary for generalized concepts of triangulations and associahedra
Simplicial complex generated by  Polytopal realization of the dual 

Triangulations (classical)  
Multiassociahedron (existence conjectured)  
Multiassociahedron of type B (existence conjectured)  
Multiclusters (present paper)  Generalized multiassociahedron (existence conjectured) 
6.1 Generalized associahedra
We have seen that for k=1, the multicluster complex \(\varDelta _{c}^{1}(W)\) is isomorphic to the ccluster complex. S. Fomin and A. Zelevinsky conjectured the existence of polytopal realizations of the cluster complex in [20, Conjecture 1.12]. F. Chapoton, S. Fomin, and A. Zelevinsky then proved this conjecture by providing explicit inequalities for the defining hyperplanes of generalized associahedra [12]. N. Reading constructed cCambrian fans, which are complete simplicial fans coarsening the Coxeter fan, see [47]. In [50], N. Reading and D. Speyer prove that these fans are combinatorially isomorphic to the normal fan of the polytopal realization in [12]. C. Hohlweg, C. Lange, and H. Thomas then provided a family of cgeneralized associahedra having cCambrian fans as normal fans by removing certain hyperplanes from the permutahedron [27]. V. Pilaud and C. Stump recovered cgeneralized associahedra by giving explicit vertex and hyperplane descriptions purely in terms of the subword complex approach introduced in the present paper [45].
6.2 Multiassociahedra of type A
 (a)
A generalized construction of the polytope of pseudotriangulations [53] using rigidity of pseudotriangulations [42, Sect. 4.2 and Remark 4.82].
 (b)
A generalized construction of the secondary polytope. As presented in [22], the secondary polytope of a point configuration can be generalized using star polygons [42, Sect. 4.3].
 (c)
The brick polytope of a sorting network [46]. This new approach brought up a large family of spherical subword complexes that are realizable as the boundary of a polytope. In particular, it provides a new perspective on generalized associahedra [45]. Unfortunately, this polytope fails to realize the multiassociahedron.
6.3 Multiassociahedra of type B
We start by proving Theorem 2.10, which says that the multicluster complex \(\varDelta ^{k}_{c}(B_{mk})\) is isomorphic to the simplicial complex of centrally symmetric ktriangulations of a regular convex 2mgon. This simplicial complex was studied in [51, 63]. We then present what is known about its polytopality. The new approach using subword complexes provides in particular very simple proofs of Corollaries 2.11, 2.13, and 2.14.
Proof of Theorem 2.10
Let S={s _{0},s _{1},…,s _{ m−k−1}} be the generators of B _{ m−k }, where s _{0} is the generator such that (s _{0} s _{1})^{4}=1∈W, and the other generators satisfy the same relations as in type A _{ m−k−1}. Then, embed the group B _{ m−k } in the group A _{2(m−k)−1} by the standard folding technique: replace s _{0} by \(s'_{mk}\) and s _{ i } by \(s'_{mk+i} s'_{mki}\) for 1≤i≤m−k−1, where the set \(S'=\{s_{1}',\dots, s_{2(mk)1}'\}\) generates the group A _{2(m−k)−1}. The multicluster complex \(\varDelta ^{k}_{c}(B_{mk})\) now has an embedding into the multicluster complex \(\varDelta ^{k}_{c'}(A_{2(mk)1})\), where c′ is the Coxeter element of type A _{2(m−k)−1} corresponding to c in B _{ m−k }; the corresponding subcomplex has the property that 2(m−k) generators (all of them except \(s'_{mk}\)) always come in pairs. Using the correspondence between ktriangulations and facets of the multicluster complex described in Sect. 2.4, the facets of \(\varDelta ^{k}_{c}(B_{mk})\) considered in \(\varDelta ^{k}_{c'}(A_{2(mk)1})\) correspond to centrally symmetric multitriangulations. □
6.4 Generalized multiassociahedra of rank 2
We now prove that multicluster complexes of rank 2 can be realized as boundary complexes of cyclic polytopes. In other words, we show the existence of rank 2 multiassociahedra. This particular case was known independently by D. Armstrong.^{4}
Theorem 6.1
(Type I _{2}(m) multiassociahedra)
The multicluster complex \(\varDelta ^{k}_{c}(I_{2}(m))\) is isomorphic to the boundary complex of a 2kdimensional cyclic polytope on 2k+m vertices. The multiassociahedron of type I _{2}(m) is the simple polytope given by the dual of a 2kdimensional cyclic polytope on 2k+m vertices.
Proof
This is obtained by Gale’s evenness criterion on the word Q=(a,b,a,b,a,…) of length 2k+m: Let F be a facet of \(\varDelta ^{k}_{c}(I_{2}(m))\), and take two consecutive letters x and y in the complement of F. Since the complement of F is a reduced expression of w _{∘}, x and y must represent different generators. Since the letters in Q are alternating, this implies that the number of letters between x and y is even. □
6.5 Generalized multiassociahedra
Recall from Sect. 2.2 that a subword complex Δ(Q,π) is homeomorphic to a sphere if and only if the Demazure product δ(Q)=π, and to a ball otherwise. This motivates the question whether spherical subword complexes can be realized as boundary complexes of polytopes [34, Question 6.4.]. We show that it is enough to consider multicluster complexes to prove polytopality for all spherical subword complexes, and we characterize simplicial spheres that can be realized as subword complexes in terms of faces of multicluster complexes.
Lemma 6.2
Every spherical subword complex Δ(Q,w _{∘}) is the link of a face of a multicluster complex Δ(c ^{ k } w _{∘}(c),w _{∘}).
Proof
Observe that any word Q in S can be embedded as a subword of Q′=c ^{ k } w _{∘}(c) for k less than or equal to the size of Q, by assigning the ith letter of Q within the ith copy of c. Since the Demazure product δ(Q) is equal to w _{∘}, the word Q contains a reduced expression for w _{∘}. In other words, the set Q′∖Q is a face of Δ(Q′,w _{∘}). The link of this face in Δ(Q′,w _{∘}) consists of subwords of Q—viewed as a subword of Q′—whose complements contain a reduced expression of w _{∘}. This corresponds exactly to the subword complex Δ(Q,w _{∘}). □
We now prove that simplicial spheres realizable as subword complexes are links of faces of multicluster complexes.
Proof of Theorem 2.15
For any spherical subword complex Δ(Q,π), we have that the Demazure product δ(Q) equals π. By Theorem 3.7, Δ(Q,π) is isomorphic to a subword complex of the form Δ(Q′,w _{∘}). Using the previous lemma, we obtain that Δ(Q,π) is the link of a face of a multicluster complex. The other direction follows since the link of a subword (i.e., a face) of a multicluster complex is itself a subword complex, corresponding to the complement of this subword. □
Finally, we prove that the question of polytopality of spherical subword complexes is equivalent to the question of polytopality of multicluster complexes.
Proof of Corollary 2.16
On one hand, if every spherical subword complex is polytopal, then clearly every multicluster complex is polytopal. On the other hand, suppose that every spherical subword complex is polytopal. Every spherical subword complex is the link of a face of a multicluster complex. Since the link of a face of a polytope is also polytopal, Theorem 2.15 implies that every spherical subword complex is polytopal. □
7 Sorting words of the longest element and the SINproperty
In this section, we give a simple combinatorial description of the csorting words of w _{∘} and prove that a word Q coincides up to commutations with c ^{ k } w _{∘}(c) for some nonnegative integer k if and only if Q has the SINproperty as defined in Sect. 2.4. This gives us an alternative way of defining multicluster complexes in terms of words having the SINproperty. Recall the involution ψ:S→S from Sect. 4 defined by \(\psi(s) = w_{\circ}^{1} s w_{\circ}\). The sorting word of w _{∘} has the following important property.
Proposition 7.1
The sorting word w _{∘}(c) is, up to commutations, equal to a word with suffix (ψ(c _{1}),…,ψ(c _{ n })), where c=c _{1}⋯c _{ n }.
Proof
As w _{∘} has a csorting word having c=(c _{1},…,c _{ n }) as a prefix, the corollary is obtained by applying Proposition 4.3 n times. □
Given a word w in S, define the function \(\phi_{\mathbf{w}}: S \rightarrow\mathbb{N}\) given by ϕ _{ w }(s) being the number of occurrences of the letter s in w.
Theorem 7.2
Proof
Sorting words of w _{∘} have intervening neighbors, see [59, Proposition 2.1] for an equivalent formulation. Therefore, s and t alternate in w _{∘}(c), with s coming first. Thus, \(\phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s) = \phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(t)\) if and only if the last t comes after the last s. Using Proposition 7.1, this means that s appears before t in ψ(c) or equivalently ψ(s) appear before ψ(t) in c. Otherwise, the last s will appear after the last t. □
It is known that if ψ is the identity on S, or equivalently if w _{∘}=−1, then the csorting word of w _{∘} is given by \({ \mathbf {w} _{\circ}(\mathbf{c})}=\mathbf{c}^{\frac{h}{2}}\), where h denotes the Coxeter number given by the order of any Coxeter element. In the case where ψ is not the identity on S (that is, when W is of types A _{ n } (n≥2), D _{ n } (n odd), E _{6} and I _{2}(m) (m odd), see [6, Exercise 10 of Chap. 4]), the previous theorem gives a simple way to obtain the sorting words of w _{∘}.
Algorithm 7.3
 (i)
Since the Coxeter diagram is connected, one can use Theorem 7.2 to compute \(\phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s)\) for all s depending on \(m := \phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(c_{1})\).
 (ii)Using that the number of positive roots equals nh/2, one obtains m and thus all \(\phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s)\) using$$2 \cdot\sum_{s \in S} \phi_{{ \mathbf {w} _\circ (\mathbf{c})}}(s) = nh. $$
 (iii)
Using that \({ \mathbf {w} _{\circ}(\mathbf{c})}= \mathbf{c}_{K_{1}} \mathbf{c}_{K_{2}} \cdots \mathbf{c}_{K_{r}}\) where K _{ i }⊆S for 1≤i≤r and c _{ I }, with I⊆S, is the Coxeter element of W _{ I } obtained from c by keeping only letters in I, we obtain that \(\mathbf{c}_{K_{i}}\) is the product of all s for which \(\phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s) \geq i\).
This algorithm provides an explicit description of the sorting words of the longest element w _{∘} of any finite Coxeter group using nothing else than Coxeter group theory. This answers a question raised in [27, Remark 2.3] and simplifies a step in the construction of the cgeneralized associahedron. We now give two examples of how to use this algorithm.
Example 7.4
Let W=A _{4} and S={s _{1},s _{2},s _{3},s _{4}} with the labeling of the graph shown in Fig. 1 on page 26. Moreover, let c=s _{1} s _{3} s _{2} s _{4}. Fix \(\phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s_{1})=m\). Since s _{1} comes before s _{2} in c and ψ(s _{1})=s _{4} comes after ψ(s _{2})=s _{3}, the letter s _{1} appears one more time than the letter s _{2} in w _{∘}(c), i.e., \(\phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s_{2})=m1\). Repeating the same argument gives \(\phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s_{3}) = m \text{ and } \phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s_{4}) = m1\). Summing up these values gives the equality \(4m2=\frac{n\cdot h}{2}=\frac{4\cdot5}{2}=10\), and thus m=3. Finally, the csorting word is w _{∘}(c)=(s _{1},s _{3},s _{2},s _{4}s _{1},s _{3},s _{2},s _{4}s _{1},s _{3}).
Example 7.5
Let W=E _{6} and S={s _{1},s _{2},…,s _{6}} with the labeling of the graph shown in Fig. 1 on page 26. Moreover, let c=s _{3} s _{5} s _{4} s _{6} s _{2} s _{1}. Fix \(\phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s_{6})=m\). Repeating the same procedure from the previous example and using that ψ(s _{6})=s _{6}, ψ(s _{3})=s _{3}, ψ(s _{2})=s _{5}, ψ(s _{1})=s _{4}, we get \(\phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s_{1}) = \phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s_{2}) = {m1}\), \(\phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s_{3}) = \phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s_{6}) = m\), \(\phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s_{4}) = \phi_{{ \mathbf {w} _{\circ}(\mathbf{c})}}(s_{5}) = m+1\). As the sum equals \(\frac{nh}{2} = \frac{6\cdot12}{2} = 36\), we obtain m=6. Finally, the csorting word is (c ^{5}s _{3},s _{5},s _{4},s _{6}s _{5},s _{4}).
Remark 7.6
We are now in the position to prove Theorem 2.7.
Proof of Theorem 2.7
Suppose that a word Q has the SINproperty; then it has complete support by definition, and it contains, up to commutations, some word c=(c _{1},…,c _{ n }) for a Coxeter element c as a prefix. Moreover, the word (ψ(c _{1}),…,ψ(c _{ n })) is a suffix of Q, up to commutations. Observe that a word has intervening neighbors if and only if it is a prefix of c ^{∞} up to commutations, see [15, Sect. 3]. In view of Lemma 4.1 and the equality δ(Q)=w _{∘}, the word Q has, up to commutations, w _{∘}(c) as a prefix. If the length of Q equals w _{∘}, the proof ends here with k=0. Otherwise, the analogous argument for \(\operatorname{rev}(Q)\) gives that the word \(\operatorname{rev}(Q)\) has, up to commutations, \(\mathbf {w} _{\circ}(\psi(\operatorname{rev}(\mathbf{c})))\) as a prefix. By Remark 7.6, the word \(\mathbf {w} _{\circ}(\psi (\operatorname{rev}(\mathbf{c})))\) is, up commutations, equal to the reverse of w _{∘}(c). Therefore, Q has the word w _{∘}(c) also as a suffix. Since c=(c _{1},…,c _{ n }) is a prefix of Q and of w _{∘}(c), and Q has intervening neighbors, Q coincides with c ^{ k } w _{∘}(c) up to commutations. Moreover, if Q is equal to c ^{ k } w _{∘}(c) up to commutations, it has intervening neighbors, and a suffix (ψ(c _{1}),…,ψ(c _{ n })), up to commutations, by Proposition 7.1. This implies that the word Q has the SINproperty. □
Remark 7.7
In light of Theorem 2.7 and Sect. 3.2, starting with a word Q having the SINproperty suffices to construct a multicluster complex, and choosing a particular connected subword in the biinfinite word \(\widetilde {Q}\), defined in Sect. 3.2, corresponds to choosing a particular Coxeter element.
We finish this section with a simple observation on the biinfinite word \(\widetilde{Q}\). For any letter q in the word Qψ(Q), let β _{ q } be the root obtained by applying the prefix w _{ q } of Qψ(Q) before q to the simple root α _{ q }. To obtain roots for all letters in \(\widetilde {Q}\), repeat this association periodically.
Proposition 7.8
Proof
8 Multicluster complexes, Auslander–Reiten quivers, and repetition quivers
In this section, we connect multicluster complexes to Auslander–Reiten quivers and repetition quivers. This approach emphasizes that the multicluster complex can be seen as a cyclic object that does not depend on a particular choice of a Coxeter element. In type A, this approach corresponds to considering subword complexes on a Möbius strip, see [43]. We use this connection to introduce an action on vertices and facets of multicluster complexes generalizing the natural rotation action on multitriangulations in type A.
Auslander–Reiten and repetition quivers play a crucial role in Auslander–Reiten theory, which studies the representation theory of Artinian rings and quivers. The Auslander–Reiten quiver Γ _{ Ω } of a quiver Ω encodes the irreducible morphisms between isomorphism classes of indecomposable representations of right modules over k Ω. These were introduced by M. Auslander and I. Reiten in [2, 3]. We also refer to [5, 21] for further background. We use these connections to describe a natural cyclic action on multicluster complexes generalizing the rotation of the polygon in types A and B, see Theorems 8.8 and 8.10.
8.1 The Auslander–Reiten quiver
In types A, D and E, sorting words of w _{∘} are intimately related to AuslanderReiten quivers. Starting with a quiver Ω _{ c } associated to a Coxeter element c (as described in Sect. 2), one can construct combinatorially the Auslander–Reiten quiver \(\varGamma_{\varOmega_{c}}\), see [7, Sect. 2.6]. R. Bédard then shows how to obtain all reduced expressions for w _{∘} adapted to Ω _{ c } (i.e., the words equal to w _{∘}(c) up to commutations) using the Auslander–Reiten quiver and a certain tableau. K. Igusa and R. Schiffler use these connections in order to obtain their description of cw _{∘}(c), see [29, Sects. 2.1–2.3]. Conversely, given the csorting word w _{∘}(c), one can recover the Auslander–Reiten quiver \(\varGamma_{\varOmega_{c}}\), see [64, Proposition 1.2] and the discussion preceding it. Algorithm 7.3 thus provides a way to construct the Auslander–Reiten quiver in finite types using only Coxeter group theory; it uses results on admissible sequences [59] and words with intervening neighbors [16].
Algorithm 8.1
 (iv)
The vertices of \(\varGamma_{\varOmega_{c}}\) are the letters of w _{∘}(c) and two letters q,q′ of w _{∘}(c) are linked by an arrow q→q′ in \(\varGamma_{\varOmega_{c}}\) if and only if q and q′ are neighbors in the Coxeter graph and q comes directly before q′ in w _{∘}(c) when restricted to the letters q and q′.
Figure 1 shows two examples of Auslander–Reiten quivers and how to obtain it using this algorithm.
8.2 The repetition quiver
Next, we define the repetition quiver.
Definition 8.2
[33, Sect. 2.2]
The repetition quiver \(\mathbb{Z}\varOmega\) of a quiver Ω consists of vertices (i,v) for a vertex v of Ω and \(i \in \mathbb{Z}\). The arrows of \(\mathbb{Z}\varOmega\) are given by (i,v)→(i,v′) and (i,v′)→(i+1,v) for any arrow v→v′ in Ω.
Remark 8.3
Vertices in the Auslander–Reiten quiver correspond to (isomorphism classes of) indecomposable representations of Ω _{ c } and thus have a dimension vector attached. By the knitting algorithm, the dimension vector at a vertex V=(i,v) of \(\varGamma_{\varOmega_{c}}\) plus the dimension vector at the vertex τ(V) equals the sum of all dimension vectors at vertices V′ for which τ(V)→V′→V are arrows in \(\varGamma_{\varOmega _{c}}\), see [24, Sect. 10.2]. This procedure is intimately related to the SINproperty, which ensures that this sum is indeed over all neighbors of v. Moreover, Proposition 7.8 implies that this property holds as well for the root β _{ q } attached to a letter q in the biinfinite sequence \(\widetilde {Q}\). This yields the wellknown property that the dimension vector and the corresponding root coincide.
The following proposition describes words for the multicluster complex using the repetition quiver, the Auslander–Reiten translate, and the shift operation.
Proposition 8.4
Let Ω _{ c } be a quiver corresponding to a Coxeter element c. Words for the multicluster complex are obtained from the biinfinite word \(\widetilde{{ \mathbf {w} _{\circ}(\mathbf{c})}}\) by setting τ ^{ k }=[1]. Choosing a particular fundamental domain for this identification corresponds to choosing a particular Coxeter element. In other words, words for multicluster complexes are obtained by a choice of linear extension of a fundamental domain of the identification τ ^{ k }=[1] in the repetition quiver.
Proof
With the identification [1]V=τ ^{ k } V in the repetition quiver, a fundamental domain will consist of k copies of Ω _{ c } and one copy of the Auslander–Reiten quiver \(\varGamma_{\varOmega_{c}}\). This fundamental domain is exactly the quiver formed from the word c ^{ k } w _{∘}(c) using Algorithm 8.1. As linear extensions of this quiver correspond to words equal to c ^{ k } w _{∘}(c) up to commutations, the result follows. □
The two boxes in Fig. 2 mark two particular choices of a fundamental domain for the multicluster complex of type A _{4} with k=1 corresponding to the Coxeter elements s _{1} s _{2} s _{3} s _{4} and s _{1} s _{3} s _{2} s _{4}, respectively.
8.3 The Auslander–Reiten translate on multicluster complexes
The Auslander–Reiten translate gives a cyclic action on the vertices and facets of a multicluster complex. This action corresponds to natural actions on multitriangulations in types A and B, and is well studied in the case of cluster complexes.
Definition 8.5
Let Q=c ^{ k } w _{∘}(c). The permutation \(\varTheta: Q\ \tilde {\rightarrow}\ Q\) is given by sending a letter q _{ i }=s to the next occurrence of s in Q, if possible, and to the first occurrence of ψ(s) in Q otherwise.
Observe that in types ADE, the operation Θ corresponds to the inverse of the Auslander–Reiten translate, Θ=τ ^{−1} when considered within the repetition quiver.
Proposition 8.6
The permutation Θ induces a cyclic action on the facets of Δ(Q,w _{∘}).
Proof
By Proposition 3.9, the subword complexes Δ(Q,w _{∘}) and \(\varDelta (Q_{\stackrel {\circlearrowleft }{s}},w_{\circ})\) are isomorphic for an initial letter s in Q. Proposition 4.3 asserts that c ^{ k } w _{∘}(c) and the rotated word obtained from c ^{ k } w _{∘}(c) by rotating n times are equal up to commutations. By construction, Θ is the automorphism of Δ(Q,w _{∘}) given by inverse rotation of c. □
Example 8.7
Theorem 8.8
Proof
To obtain the order of this action, we consider the length of Q divided by the length of c if \(w_{\circ}\equiv{\bf1}\) and twice the length of Q divided by the length of c otherwise. We have already seen in Algorithm 7.3 that the length of Q is given by kn+nh/2. As the length of c is given by n, the result follows. □
Remark 8.9
The action induced by the tropical Coxeter element on facets of the cluster complex was shown by S.P. Eu and T.S. Fu to exhibit a cyclic sieving phenomenon [17]. Therefore, the cyclic action induced by Θ exhibits a cyclic sieving phenomenon for facets of the cluster complex Δ(cw _{∘}(c),w _{∘}) and any Coxeter element c.
Finally, for types A and B, the cyclic action \(\varTheta: Q \tilde {\rightarrow} Q\) corresponds to the cyclic action induced by rotation of the associated polygons.
Theorem 8.10
Let Q=c ^{ k } w _{∘}(c). In type A _{ m−2k−1}, the cyclic action Θ on letters in Q corresponds to the cyclic action induced by rotation on the set of krelevant diagonals of a convex mgon. In type B _{ m−k }, the cyclic action Θ corresponds to the cyclic action induced by rotation on the set of krelevant centrally symmetric diagonals of a regular convex 2mgon.
Proof
The simplicial complex of ktriangulations of a convex mgon is isomorphic to the multicluster complex of type A _{ m−2k−1}, so the order of Θ is given by 2k+h=2k+m−2k=m as expected. The simplicial complex of centrally symmetric ktriangulations of a regular convex 2mgon is isomorphic to the multicluster complex of type B _{ m−k }, so the order of Θ equals k+h/2=k+m−k=m as well. In type A, the result follows from the correspondence between letters in Q and krelevant diagonals in the mgon as described in Sect. 2.4. In type B, the result follows from the correspondence between letters in Q and krelevant centrally symmetric diagonals in the 2mgon as described in Sect. 2.4. □
9 Open problems
We discuss open problems and present several conjectures. We start with two open problems concerning counting formulas for multicluster complexes.
Open Problem 9.1
Find multiCatalan numbers counting the number of facets in the multicluster complex.
Open Problem 9.2
Open Problem 9.3
Construct a family of simplicial complexes which simultaneously contains generalized cluster complexes and multicluster complexes.
The next open problem concerns a possible representation theoretic description of the multicluster complex in types ADE. For k=1, one can describe the compatibility by saying that V∥_{ c } V′ if and only \(\dim(\operatorname{Ext}^{1}(V,V')) = 0\), see [9].
Open Problem 9.4
Describe the multicluster complex within the repetition quiver using similar methods.
The following problem extends the diameter problem of the associahedron to the family of multicluster complexes, see [42, Sect. 2.3.2] for further discussions in the case of multitriangulations.
Open Problem 9.5
Find the diameter of the facetadjacency graph of the multicluster complex \(\varDelta ^{k}_{c}(W)\).
Finally, we present several combinatorial conjectures on the multicluster complexes. We start with a conjecture concerning minimal nonfaces.
Conjecture 9.6
Minimal nonfaces of the multicluster complex \(\varDelta ^{k}_{c}(W)\) have cardinality k+1.
Since w _{∘} is csortable, we have \(\mathbf{c}^{k}{ \mathbf {w} _{\circ}(\mathbf{c})}= \mathbf{c}^{k}\mathbf{c}_{K_{1}}\mathbf{c}_{K_{2}}\cdots \mathbf{c}_{K_{r}}\) with K _{ r }⊆⋯⊆K _{2}⊆K _{1}. This implies that the complement of any k letters still contains a reduced expression for w _{∘}. In other words, minimal nonfaces have at least cardinality k+1. Moreover, using the connection to multitriangulations and centrally symmetric triangulations, we see that the conjecture holds in types A and B. It also holds in the case of dihedral groups: it is not hard to see that the faces of the multicluster complex are given by subwords of c ^{ k } w _{∘}(c)=(a,b,a,b,…) that do not contain k+1 pairwise nonconsecutive letters (considered cyclically). The conjecture was moreover tested for all multicluster complexes of rank 3 and 4 with k=2.
In types A and I _{2}(m), there is a binary compatibility relation on the letters of c ^{ k } w _{∘}(c) such that the faces of the multicluster complex can be described as subsets avoiding k+1 pairwise incompatible elements. We remark that this is not possible in general: in type B _{3} with k=2, as in Example 2.12, \(\varDelta _{c}^{2}(B_{3})\) is isomorphic to the simplicial complex of centrally symmetric 2triangulations of a regular convex 10gon. Every pair of elements in the set \(\mathcal{A}=\{ [1,4]_{\mathrm{sym}}, [4,7]_{\mathrm{sym}}, [7,10]_{\mathrm{sym}} \}\) is contained in a minimal nonface. But since \(\mathcal{A}\) does not contain a 3crossing, it forms a face of \(\varDelta _{c}^{2}(B_{3})\).
Theorem 2.7 gives an alternative way of defining multicluster complexes as subword complexes Δ(Q,w _{∘}) where the word Q has the SINproperty. It seems that this definition covers indeed all subword complexes isomorphic to multicluster complexes.
Conjecture 9.7
Let Q be a word in S with complete support, and π∈W. The subword complex Δ(Q,π) is isomorphic to a multicluster complex of type W if and only if Q has the SINproperty and π=δ(Q)=w _{∘}.
The fact that π=δ(Q) is indeed necessary so that the subword complex is a sphere. It remains to show that π=w _{∘} and that Q has the SINproperty. One reason for this conjecture is that if Q does not have the SINproperty, then it seems that the subword complex Δ(Q,w _{∘}) has fewer facets than required. Indeed, we conjecture that multicluster complexes maximize the number of facets among all subword complexes with a word Q of a given size.
Conjecture 9.8
Let Q be any word in S with kn+N letters (where N denotes the length of w _{∘}), and Δ(Q,w _{∘}) be the corresponding subword complex. The number of facets of Δ(Q,w _{∘}) is less than or equal to the number of facets of the multicluster complex \(\varDelta ^{k}_{c}(W)\). Moreover, if both numbers are equal, then the word Q has the SINproperty.
We remark that the previous two conjectures hold for the dihedral types I _{2}(m). In this case, the multicluster complex is isomorphic to the boundary complex of a cyclic polytope, which is a polytope that maximizes the number of facets among all polytopes in fixed dimension on a given number of vertices, see, e.g., [65]. Moreover, we present below a simple polytope theory argument in order to show that if a word does not satisfy the SINproperty, then the corresponding subword complex has strictly less facets than the multicluster complex. First note that Corollary 2.16 and Theorem 6.1 imply that all spherical subword complexes of type I _{2}(m) are polytopal. By the upper bound theorem, a polytope has as many facets as a cyclic polytope if and only if it is neighborly, see, e.g., [65]. Therefore, it is enough to prove that if Q=(q _{1},…,q _{ r }) with r=2k+m is a word in S={a,b} containing two consecutive letters that are equal, then the subword complex Δ(Q,w _{∘}) is not neighborly. Since this is a 2kdimensional complex, this is equivalent to show that there is a set of k letters of Q that do not form a face. By applying rotation of letters and Proposition 3.9, we can assume without loss of generality that the last two letters of Q are equal. Among the first 2k+1 letters of Q, one of the generators a or b appears no more than k times. The set of these no more than k letters is not a face of the subword complex. The reason is that the reduced expressions in the complement of this set in Q have length at most m−1, which is one less than the length of w _{∘}.
In view of Corollary 2.16, the following conjecture restricts the study of [34, Question 6.4].
Conjecture 9.9
The multicluster complex is the boundary complex of a simplicial polytope.
In types A and B, this conjecture coincides with the conjecture on the existence of the corresponding multiassociahedra, see [31, 63], and Theorem 6.1 shows that this conjecture is true for dihedral groups.
Footnotes
 1.
We thank an anonymous referee for pointing us to this result.
 2.
The proof appeared in Sect. 7 in the arxiv version, see http://arxiv.org/abs/0904.1097v2.
 3.
As far as we know, the first reference to this conjecture appears in [31, Sect. 1].
 4.
Personal communication.
 5.
The proof appeared in Sect. 7 in the arxiv version, see http://arxiv.org/abs/0904.1097v2.
 6.
Personal communication with V. Reiner.
Notes
Acknowledgements
The authors would like to thank Hugh Thomas for pointing them to the results in [29], and Christophe Hohlweg, Carsten Lange, Vincent Pilaud, Hugh Thomas, and Günter M. Ziegler for important comments and remarks on preliminary versions of this article. The first two authors are thankful to Drew Armstrong, Carsten Lange, Emerson Leon, Ezra Miller, Vincent Pilaud, and Luis Serrano for useful discussions that took place during the 23rd FPSAC Conference in Reykjavik. They are particularly grateful to Carsten Lange and Emerson Leon for numerous fruitful discussions that took place in the Arnimallee 2 Villa in Berlin.
We used the computer algebra system Sage [56] for implementing the discussed objects, and to test the conjectures.
C. Ceballo was supported by DFG via the Research Training Group “Methods for Discrete Structures” and the Berlin Mathematical School.
J.P. Labbé was supported by an FQRNT Doctoral scholarship.
C. Stump was partially supported by a CRMISM postdoctoral fellowship and the DFG via the Research Group “Methods for Discrete Structures”.
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