Generalized Associahedra and Noncrossing Partitions

GENERALIZED ASSOCIAHEDRA AND NONCROSSING PARTITIONS

TOM BRADY, JON MCCAMMOND, NATHAN READING, AND HUGH THOMAS

Abstract. In this article we collect together the basic definitions of seven combinatorial defined objects derived from finite Coxeter groups: noncrossing parititions, Tom Brady and Colum Watt’s root complex and Petrie complex, Sergey Fomin and Andrei Zelevinsky’s simplicial and simple associahedra, and Nathan Reading’s Lattice congruence and diminishing element poset. In addi- tion to describing what should be true about the combinatorial relationships between these objects, this file (hopefully) will soon contain proofs of the various relationships including a proof of Chapoton’s detailed numerology.

1. Introduction The origin of this article lies in the conference on “Braid groups, Clusters and Free Probability” held at the American Institute of Mathematics January 10-14, 2005. At that conference, Tom Brady and Nathan Reading both announced results which firmly connected noncrossing partitions and generalized associahedra for the first time. Numerological connections between the two have been observed for several years and were part of the reason the conference was organized. Over the years, a long list of conjectures have been produced which can now be verified.

2. Coxeter Basics This section contains background stuff on the W -permutahedron and its key properties. Let S be a simple system for W and let γ be a bipartite Coxeter element formed from S. The selection of S corresponds to a choice of a vertex of P . There is a natural height function on the W -permutahedron that turns its 1- skeleton into a directed graph. Moreover, the edges in the 1-skeleton have two natural labelings; one by the elements of S and one by the elements of T , the full set of reflections. Definition 2.1 (W -permutahedra). Definition 2.2 (S and T labelings). If the action of W on Rn is a left action, then the natural S-labeling of the 1-skeleton of Perm(W ) is the right Cayley graph of W with respect to S and the natural T -labeling of the 1-skeleton is a portion of the left Cayley graph of W with respect to T . Root systems, and Minkowski sums Coxeter element Coxeter plane Necklace of simplices

Date: March 16, 2005. 1 GENERALIZED ASSOCIAHEDRA AND NONCROSSING PARTITIONS 2

Petrie polygon

a = (12) b 2 3 b = (23) f c = (34) a c e d = (14) 1 4 e = (13) d f = (24)

Figure 1. Standard notations for Type A3

3. The Objects 3.1. Generalized Associahedra. Generalized associahedra were initially defined by Sergey Fomin and Andrei Zelevinsky in [cite]. The object that natural arose from their work on cluster algebras was a graph where the associated flag complex was a simplicial subdivision of a sphere. What was not originally obvious was whether these simplicial complexes were boundary complexes of Euclidean polytopes. That this was indeed the case was proved latter, by explicitly writing down the equations that needed to be satisfied and showing that this system had a nontrivial solution. The approach we take here is somewhat different. We begin with a naturally defined doubly-labeled convex polytope that is associated with the almost positive roots and then we later prove that this labeled polytope is indeed the W -associahedron as constructed by Fomin and Zelevinsky. Introduce Generalized Associahedra and mention Cluster algebras.

3.2. Noncrossing partitions. Introduce factor posets, and mention Garside struc- tures. Let σ be a permutation in Symn with exactly k cycles when written in disjoint cycle notation. Trivial cycles are included in this count. Multiplying σ by a 2-cycle τ = (ij) can at most reduce the number of cycles by 1. More precisely, στ has k −1 cycles when i and j belong to distinct cylces of σ and k + 1 cycles when i and j lie in the same cycle of σ. From these observations it is easy to see that σ cannot be written as a product of fewer than n−k 2-cycles (since n is the number of cycles for the identity element) and it is also easy to produce such a factorization by starting with σ and repeatedly multiplying by 2-cycles involving two elements that remain in the same cycle at that stage.

3.3. Lattice congruences. Introduce Nathan’s lattice congruences based on a choice of Coxeter elements and mention quiver representations. Also deﬁne the diminishing elements stuﬀ.

3.4. Nonnesting partitions. Introduce nonnesting partitions and mention the Borel ad-Nil ideals Root poset GENERALIZED ASSOCIAHEDRA AND NONCROSSING PARTITIONS 3

a −c b

e f −e

−a d c

Figure 2. The A3 cluster complex

a −c b

Te f B −e

−a d c

Figure 3. The A3 cluster complex, doubly labeled GENERALIZED ASSOCIAHEDRA AND NONCROSSING PARTITIONS 4

b a −c

e f −e

−a d c

Figure 4. The dual A3 complex

Edges Facets Partitions in NCW ⇔ Vertices of Dual(GAW ) ⇔ Type W -Clusters

Figure 5. The overview for simple generalized associahedra

4. The Maps Give the proof scheme and deﬁne all of the relevant maps.

5. The Proofs There are two pictures with natural labelings which can be superimposed.

Walls Vertices Partitions in NCW ⇔ Facets of GAW ⇔ Type W -Clusters

Figure 6. The overview for simplicial generalized associahedra

6. Coxeter basics Definition 6.1 (Root systems). Classification. Definition 6.2 (W -permutahedron). Two different labelings. Coxeter elements, exponents, Theorem 6.3. For each finite Coxeter group W , the Coxeter elements of W form a single conjugacy class. Theorem 6.4 (Tit’s solution to the word problem). GENERALIZED ASSOCIAHEDRA AND NONCROSSING PARTITIONS 5

7. Quivers and Coxeter elements Lemma 7.1 (Factorizations and paths). Let W be an irreducible rank n Coxeter group and let P be the standard reflection labeled W -permutahedron. There is a natural 2 to 1 map between directed paths in P of length n which are not contained in any facet and the words of length n in T ∗ which represent some Coxeter element of W . Lemma 7.2. Forgetting the path but remembering the endpoints corresponds to a chamber with a quiver The first step is to classify which chambers can be used to define a particular Coxeter element γ. Theorem 7.3. Let W be a finite Coxeter group and let γ be a fixed Coxeter element. The chambers which can be used to define γ are in 2 − 1 correspondence with the labeled W -quivers.

W {Maximal chains in NC / ∼c.e } m {Labeled metric W -associahedra} m {Labeled W -quivers}

Figure 7

W {γ-equivalence classes of maximal chains in NC / ∼c.e} m {Metric W -associahedra} m {W -quivers}

Figure 8

8. The Double Labeling There are two pictures with natural labelings which can be superimposed. (yield- ing the double labeling)

Edges Facets Partitions in NCW ⇔ Vertices of Dual(GAW ) ⇔ Type W -Clusters

Figure 9. The overview for simple generalized associahedra

Walls Vertices Partitions in NCW ⇔ Facets of GAW ⇔ Type W -Clusters

Figure 10. The overview for simplicial generalized associahedra GENERALIZED ASSOCIAHEDRA AND NONCROSSING PARTITIONS 6

a −c b

e f −e

−a d c

Figure 11. The A3 cluster complex

9. The cubical core Let W be a ﬁnite irreducible Coxeter group and let γ be a ﬁxed Coxeter element for W . The cubical core of the W -permutahedron relative to the γ-action is the cube complex containing the geodesic loops commutation equivalent to the loop γn.

−A

−B

10. Polytopal realizations of the associahedron There are various polytopal realizations of the generalized associahedra. Chapo- ton, Fomin, and Zelevinsky [1]. Ours is based on removing facets of the W -permutahedron. There is an intermediate version that Hugh and I have been calling the “top”. Keep only those 2N facets corresponding to the roots as labeled by the double labeling. The Top has a some of the original vertices of the permutahedron but not many. Those that remain all seem to lie on the “cubical core” of W with respect to γ. GENERALIZED ASSOCIAHEDRA AND NONCROSSING PARTITIONS 7

−A

Theorem 10.1. We hopefully can show that for every Dynkin quiver there is a metric realization of the generalized associahedron. Proof. The idea of the proof is to remove certain facets from the W -permutahedron. First remove those that are unlabeled. Then pick a pair of antipodal original vertices. Only keep those facets that are “between” them in the cyclic root ordering.

Corollary 10.2. For every metric generalized associahedron there is a natural bijection to the noncrossing partition NCW and a natural height function.

11. Late breaking news Here are some recent insights. Definition 11.1 (Chamber core). Let W be a finite Coxeter group and let γ be one of its Coxeter elements. For each reflection r ∈ T there is a unique S0 fixed by γ · r. Let V denote the collection of these vertices. Because γ is a Coxeter element there is exists at least one chamber supported by this vertex set. In fact, the set of all chambers supported by this vertex sets corresponds exactly to the set of all chambers / simple systems whose reflections can be multiplied together in some order to produce γ. In other words these are the chambers that produce chamber factorizations of γ. The smallest subcomplex of the Coxeter complex containing all these chambers is what we call the chamber core of (W, γ). The structure of the chamber core is quite simple. It is homotopy equivalent to a circle. It has a natural orientation coming from γ. It is γ-invariant. Definition 11.2 (Cubical core). The internal dual of the chamber core is what we call the cubical core of (W, γ). Its structure is even simpler. It is a cube complex contained in the W -permutahedron. It is also homotopy equivalent to a circle. It’s edges have a natural orientation coming from γ, giving rise to a natural cyclic partial ordering of its vertex set. GENERALIZED ASSOCIAHEDRA AND NONCROSSING PARTITIONS 8

The symmetric group of this edge oriented cube complex can be quite large and the equivalence classes of the vertex set are in 1-1 correspondence with the quivers of the Dynkin diagram. The edges correspond to what representation theorists call reflection functors. Definition 11.3 (The cyclic ordering). Let C be any complex which is h.e. to a circle, has an oriented 1-skeleton, has antipodal symmetries (so that −v makes sense), and where all positively oriented loops have positive winding number. We say that v>u if there is a directed loop with winding number 1 which contains, in order v, u, −v, and −u. Thus, v>u if and only if u > −v, −v > −u and −u>v. In the same way that standard posets can be linearized, cyclic posets can be circularized. (make this precise) Definition 11.4 (Metric associahedra). Let W be a finite Coxeter group, let S be a simple system for W and let γ be a product of S in some order. Let C be the interval the chamber complex between the two chambers labeled by S. From the intersection of halfspaces forming the W -permutahedron, select out those which correspond to the vertices of C. Combinatorially this will be a W -associahedron and it will have a metric structure coming from the particular quiver structure (γ,S). We know this is a bounded polytope because intersecting just the halfspaces which correspond to the vertices of the two opposite chambers gives a parallelop- iped. GENERALIZED ASSOCIAHEDRA AND NONCROSSING PARTITIONS 9

Appendix A. Coxeter numerology

Type #Quivers A2 1 A3 2 A4 3 D4 2 A5 6 D5 6 A6 10 D6 12 E6 10 A7 20 D7 24 E7 64 A8 36 D8 48 E8 128 A9 72 D9 96 m−2 m−1 A2m 2 (2 + 1) m−1 m−1 A2m+1 2 (2 + 1) n−4 Dn(n > 4) 3 · 2

Type h 2N N An n − 1 n(n − 1) n(n − 1)/2 Dn 2(n − 1) 2n(n − 1) n(n − 1) 2 2 Bn 2n 2n n I2(m) m 2m m

Figure 12

Type Exponents Degrees Size An 1, 2,...,n − 1 2, 3,...,n n! n−1 Dn 1, 3,..., 2n − 3, n − 1 2, 4,..., 2n − 2, n 2 · n! n Bn 1, 3,..., 2n − 3, 2n − 1 2, 4,..., 2n − 2, 2n 2 · n! I2(m) 1,m − 1 2,m 2m

Figure 13

References

[1] Fr´ed´eric Chapoton, Sergey Fomin, and Andrei Zelevinsky. Polytopal realizations of generalized associahedra. Canad. Math. Bull., 45(4):537–566, 2002. Dedicated to Robert V. Moody. GENERALIZED ASSOCIAHEDRA AND NONCROSSING PARTITIONS 10

Type Mu Catalan Narayana A2 2 5 1-3-1 B2 3 6 1-4-1 A3 -5 14 1-6-6-1 B3 -10 20 1-9-9-1 H3 -21 32 1-15-15-1 A4 14 42 1-10-20-10-1 D4 20 50 1-12-24-12-1 B4 35 70 1-16-36-16-1 F4 66 105 1-24-55-24-1 H4 232 280 1-60-158-60-1 A5 -42 132 1-15-50-50-15-1 D5 -77 182 1-20-70-70-20-1 B5 -126 252 1-25-100-100-25-1 A6 132 429 1-21-105-175-105-21-1 D6 294 672 1-30-165-280-165-30-1 B6 462 924 1-36-225-400-225-36-1 E6 418 833 1-36-204-351-204-36-1 A7 -429 1430 1-28-196-490-490-196-28-1 D7 xxx 2508 1-42-336-875-875-336-42-1 B7 -1716 3432 1-49-441-1225-1225-441-49-1 E7 -2431 4160 1-63-546-1470-1470-546-63-1 A8 1430 4862 1-36-336-1176-1764-1176-336-36-1 D8 xxx 9438 1-56-616-2296-3500-2296-616-56-1 B8 6435 12870 1-64-784-3136-4900-3136-784-64-1 E8 xxx 25080 1-120-xxx-xxx-xxx-xxx-xxx-120-1 A9 -4862 16796 1-45-540-2520-5292-5292-2520-540-45-1 D9 xxx 35750 1-72-xxx-xxx-xxx-xxx-xxx-72-1 B9 -24310 48620 1-81-1296-7056-15876-15876-7056-1296-81-1 Type Mu Catalan Narayana

n (−1) 2n 1 2n+2 1 n n n A n+1 n n+2 n+1 n kk+1

2n 2n−2 n D ??? n − n−1 ???

n 2n−1 2n n 2 Bn (−1) n n k

I2(m) m − 1 m + 2 1-m-1

di+h W Q di