Brick polytopes, lattices and Hopf algebras

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 Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and words U,V1,...,Vk,W on [n], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytope


The motivation of this paper comes from relevant combinatorial, geometric, and algebraic structures on permutations, binary trees and binary sequences. Classical surjections from permutations to binary trees (BST insertion) and from binary trees to binary sequences (canopy) yield: • lattice morphisms from the weak order, via the Tamari lattice, to the boolean lattice; • normal fan coarsenings from the permutahedron, via J.-L. Loday's associahedron [Lod04], to the parallelepiped generated by the simple roots e i+1 − e i ; • Hopf algebra refinements from C. Malvenuto and C. Reutenauer's algebra [MR95], via J.-L. Loday and M. Ronco's algebra [LR98], to L. Solomon's descent algebra [Sol76]. These fascinating connections were widely extended by N. Reading in his work on Lattice congruences, fans and Hopf algebras [Rea05]. He proves that any lattice congruence ≡ of the weak order on S n defines a complete simplicial fan F ≡ refined by the Coxeter fan, and he characterizes in terms of simple rewriting rules the families (≡ n ) n∈N of lattice congruences of the weak orders on (S n ) n∈N which yield Hopf subalgebras of C. Malvenuto and C. Reutenauer's algebra on permutations [MR95]. His work opens two natural questions. On the geometric side, it is not clear which of the fans F ≡ are actually normal fans of polytopes, as in the previous example of the associahedron. On the algebraic side, this construction produces a combinatorial Hopf algebra whose basis is indexed by the congruence classes of (≡ n ) n∈N . However, the product and coproduct in this Hopf algebra are performed extrinsically: the algebra is embbeded in C. Malvenuto and C. Reutenauer's algebra on permutations and the computations are performed at that level. The remaining challenge is to realize the resulting Hopf algebra intrinsically by attaching a combinatorial object to each congruence class of (≡ n ) n∈N and working out the rules for product and coproduct directly on these combinatorial objects.
The present paper answers these two questions for a relevant family of lattice congruences of the weak order, generalizing the classical sylvester congruence [HNT05]. Our starting point is the world of acyclic multitriangulations [PS09,PS12]. Using classical point-line duality, it was shown in [PP12] that acyclic k-triangulations of the (n + 2k)-gon can be equivalently modeled by acyclic pipe dreams [KM05,BB93] for the permutation (1, . . . , k, n + k, . . . , k + 1, n + k + 1, . . . , n + 2k). In this paper, we call these pipe dreams acyclic (k, n)-twists. On the combinatorial side (Section 1), we provide a clean description of the natural surjection ins k from the permutations of S n to the acyclic (k, n)-twists. The fibers of this map coincide with the classes of the congruence on S n defined as the transitive closure of the rewriting rule . This shows that the increasing flip poset on the acyclic (k, n)-twists is a lattice, and provides lattice homomorphisms from the weak order and to the lattice of k-recoils classes [NRT11]. On the geometric side (Section 2), we briefly recall the definition of the brick polytope, whose vertices correspond to the acyclic (k, n)-twists. The inclusions between the permutahedron, the brick polytope, and a well-chosen graphical zonotope provide a geometric interpretation of the lattice homomorphisms mentioned above. Finally, on the algebraic side (Section 3), the congruence ≡ k defines a Hopf subalgebra of C. Malvenuto and C. Reutenauer's algebra on permutations [MR95], whose product and coproduct we interpret in terms of acyclic (k, n)-twists. Interestingly, the product in this algebra can be naturally decomposed into 2 k distinct associative operations, motivating the introduction of k-twistiform operads [HP15].
The results presented here significantly extend in three independent directions: to the Cambrian setting (where the shape is not triangular anymore), to the tuple setting (where we consider simultaneously several twists, e.g. twin acyclic (k, n)-twists), and to the Schröder setting (where we simultaneously all faces of the brick polytope). Due to space limitations, these extensions as well as all proofs of our results are skipped in this extended abstract. For more detailed information, the reader is invited to consult the long version of this paper [Pil15].
1 Combinatorics of acyclic twists 1.1 (k, n)-twists A pipe dream is a filling of a triangular shape with crosses and elbows so that all pipes entering on the left side exit on the top side. These objects were studied in the literature, under different names including "pipe dreams" [KM05], "RC-graphs" [BB93], or as specific "pseudoline arrangements on sorting networks" [PS12]. This paper is concerned with the following special family of pipe dreams.  Definition 1 For k, n ∈ N, a (k, n)-twist is a pipe dream with n + 2k pipes such that • no two pipes cross twice (the pipe dream is reduced), • the pipe which enters in row i exists in column i if k ≤ i ≤ n + k, and n + 2k + 1 − i otherwise. Besides the first k and last k trivial pipes, a (k, n)-twist has n relevant pipes, labeled by [n] from bottom to top, or equivalently from left to right (see Figure 1). We denote by T k (n) the set of (k, n)-twists.

Definition 2
The contact graph of a (k, n)-twist T is the directed multigraph T # with vertex set [n] and with an arc from the SE-pipe to the WN-pipe of each elbow in T involving two relevant pipes (see Figure 1). We say that a twist T is acyclic if its contact graph T # is (no oriented cycle), and we then denote by T the transitive closure of T # . We denote by AT k (n) the set of acyclic (k, n)-twists.
Remark 3 A k-triangulation of a convex (n + 2k)-gon is a maximal set of diagonals of the polygon such that no k +1 of them are pairwise crossing. Multitriangulations appeared in the context of extremal theory for geometric graphs and were studied for their rich combinatorial structure, see references in [PS09]. Based on k-stars in k-triangulations [PS09], it was shown in [PP12] that the map sending an elbow in row i and column j of the (n + 2k) × (n + 2k) triangular shape to the diagonal [i, j] of a (n + 2k)-gon gives a bijection between the (k, n)-twists and the k-triangulations of the (n + 2k)-gon (see Figure 2).
Example 4 For k = 1, a (1, n)-twist T corresponds to a triangulation T of a convex (n + 2)-gon, and the contact graph of T coincides with the dual binary tree of T .
Remark 5 It is known [Jon05] that the number of k-triangulations of the (n + 2k)-gon is the Hankel determinant of Catalan numbers det(C n+2k−i−j ) i,j∈ [k] , see [OEI10,A078920]. To our knowledge, there is no enumerative result on acyclic k-triangulations. See [OEI10, A263791] for the first values of |AT k (n)|.

k-twist correspondence
We present a surjection from permutations to acyclic twists. It relies on an insertion operation on twists similar to the insertion in binary search trees. Given a permutation τ := τ 1 . . . τ n , we start from the empty triangular shape and insert the pipes τ n , . . . , τ 1 such that each new pipe is as northwest as possible in the space left by the pipes already inserted (see Figure 3). We denote by ins k (τ ) the resulting (k, n)-twist.
Proposition 6 For any (k, n)-twist T, the permutations τ ∈ S n such that ins k (τ ) = T are precisely the linear extensions of the contact graph of T. In particular, ins k is surjective from S n to AT k (n).
Remembering the insertion order, ins k defines a bijection from permutations of S n to leveled acyclic (k, n)-twists, i.e. (k, n)-twists whose pipes have been relabeled by [n] so that the label of the SE-pipe is larger than the label of the WN-pipe at each elbow. We call this bijection the k-twist correspondence.
Example 7 For k = 1, the contact graph of the 1-twist ins 1 (τ ) is the binary search tree obtained by the successive insertions of the entries of τ read from right to left. If we additionally remember the insertion order, we obtain the sylvester correspondence [HNT05] between permutations and leveled binary trees.

k-twist congruence
We now characterize the fibers of ins k as classes of a congruence ≡ k on S n defined by a simple rewriting rule, similar to the sylvester congruence [HNT05].
Definition 8 The k-twist congruence ≡ k on S n is obtained as the transitive closure of the rewriting rule Proposition 9 For any τ, τ ∈ S n , we have τ ≡ k τ ⇐⇒ ins k (τ ) = ins k (τ ).
Example 10 For k = 1, the 1-twist congruence is precisely the sylvester congruence defined in [HNT05].
Remember that the (right) weak order on S n is defined as the inclusion order of coinversions, where a coinversion of τ ∈ S n is a pair of values i, j ∈ N such that i < j while τ −1 (i) > τ −1 (j). Let us also recall that a lattice congruence is an equivalence relation ≡ on a lattice (L, ∧, ∨) compatible with meets and joins: for any x ≡ x and y ≡ y , we have x ∧ y ≡ x ∧ y and x ∨ y ≡ x ∨ y . The congruence classes L/≡ then inherit a lattice structure: the order relation is defined by X ≤ Y in L/≡ if and only if there exists representatives x ∈ X and y ∈ Y such that x ≤ y in L, and the meet X ∧ Y (resp. the join X ∨ Y ) of two congruence classes X and Y is the congruence classe of x ∧ y (resp. of x ∨ y) for arbitrary representatives x ∈ X and y ∈ Y .
Proposition 11 The k-twist congruence ≡ k is a lattice congruence of the weak order.

Increasing flip lattice
We now recall the notion of flips in pipe dreams and study the graph of increasing flips in acyclic k-twists.
Definition 12 A flip in a k-twist is the exchange of an elbow between two relevant pipes with the unique crossing between them. The flip is increasing if the initial elbow is located (largely) south-west of the final elbow. The increasing flip order is the transitive closure of the increasing flip graph on AT k (n). Remark 13 An elbow flip in a k-twist T corresponds to a diagonal flip in its dual k-triangulation T [PS09]. The polygon in Figure 2 is chosen so that increasing flips in T correspond to slope increasing flips in T .
Example 14 For k = 1, the increasing flip order is isomorphic to the Tamari lattice on binary trees.
The k-twist correspondence transports the lattice structure on (S n , ≤)/≡ k to acyclic k-twists.

Proposition 15
The following posets are all isomorphic: • the increasing flip order on acyclic k-twists, • the quotient lattice of the weak order by the k-twist congruence ≡ k , • the subposet of the weak order induced by the permutations of S n avoiding the generalized pattern 1(k + 2) -(σ 1 + 1) -· · · -(σ k + 1) for any σ ∈ S k .

k-recoils and k-canopy schemes
To prepare the definition of the canopy of acyclic k-twists, we first recall the notion of k-recoil schemes of permutations already considered in [NRT11]. We use a description in terms of acyclic orientations as it is closer to the description of the vertices of the zonotope that we will use later in Section 2. We consider the graph G k (n) with vertex set [n] and edge set {i, j} ∈ [n] 2 i < j ≤ i + k (see Figure 5). We denote by AO k (n) the set of acyclic orientations of G k (n), i.e. with no oriented cycle. Definition 16 The k-recoil scheme of a permutation τ ∈ S n is the orientation rec k (τ ) ∈ AO k (n) with an edge i → j for all i, j ∈ [n] such that |i−j| ≤ k and τ −1 (i) < τ −1 (j). The map rec k : S n → AO k (n) is called k-recoil map. The k-recoil congruence ≈ k on S n is the transitive closure of the rewriting rule U ijV ≈ k U jiV , where i, j ∈ [n] are such that i + k < j while U, V are (possibly empty) words on [n].
Definition 19 A flip in an acyclic orientation of G k (n) is the switch of the direction of an edge of G k (n). The flip is increasing if the initial direction was increasing. The increasing flip order on AO k (n) is the transitive closure of the increasing flip graph on AO k (n).
Theorem 20 The k-recoil congruence ≈ k is a lattice congruence of the weak order. The k-recoil map rec k defines an isomorphism from the quotient lattice of the weak order by the k-recoil congruence ≈ k to the increasing flip lattice on the acyclic orientations of G k (n).
We can now generalize the notion of canopy of binary trees to acyclic twists. To ensure that Definition 22 is valid, we need the following observation on comparisons of closed pipes in a k-twist.
Lemma 21 If |i − j| ≤ k, the ith and jth pipes in an acyclic k-twist T are comparable for T .

Definition 22
The k-canopy scheme of a (k, n)-twist T is the orientation can k (T) ∈ AO k (n) with an edge i → j for all i, j ∈ [n] such that |i − j| ≤ k and i T j. It indeed defines an acyclic orientation of G k (n) by Lemma 21. We call k-canopy the map can k : AT k (n) → AO k (n).
Example 23 For k = 1, the 1-canopy scheme of a (1, n)-twist T is the classical canopy of T # , i.e. the sign vector can(T) ∈ {−, +} n−1 defined by can(T) i = − if the node i of T # is above the node i + 1 of T # and can(T) i = + otherwise. This map was already used by J.-L. Loday in [LR98,Lod04], but the name "canopy" was coined by X. Viennot [Vie07]. The binary search tree insertion map and the canopy map factorize the recoil map: can • ins = rec.

Proposition 24
The maps ins k , can k , and rec k define a commutative diagram of lattice homomorphisms:

Geometry of acyclic twists
This section is devoted to the polyhedral geometry of permutations of S n , acyclic twists of AT k (n), and acyclic orientations of AO k (n). It is mainly based on properties of brick polytopes of sorting networks, defined and studied by V. Pilaud and F. Santos in [PS12].

Permutahedra, brick polytopes, and zonotopes
We denote by (e i ) i∈[n] the canonical basis of R n and let 1 1 = i∈[n] e i . We recall the definition of three families of polytopes, which are illustrated in Figure 6: Permutahedron. The permutahedron Perm(n) is the (n−1)-dimensional polytope defined as the convex hull Perm(n) := conv i∈[n] ie τ (i) | τ ∈ S n . Its vertices correspond to the permutations of S n .
Brick polytope. We call bricks the squares [i, i + 1] × [j, j + 1] of the triangular shape. The brick area of a pipe p is the number of bricks located below p but inside the axis-parallel rectangle defined by the two endpoints of p. The brick vector of a k-twist T is the vector of R n whose ith coordinate is the brick area of the ith pipe of T. The brick polytope Brick k (n) is the (n − 1)-dimensional polytope defined as the convex hull of the brick vectors of all (k, n)-twists. Its vertices correspond to the acyclic (k, n)-twists of AT k (n).
Zonotope. The graphical zonotope Zono(G) of a graph G is the Minkowski sum of the segments [e i , e j ] for all edges {i, j} of G. We consider the (n−1)-dimensional zonotope Zono k (n) := Zono(G k (n)).

Its vertices correspond to the acyclic orientations of AO k (n).
Example 25 For k = 1, the brick polytope Brick 1 (n) coincides (up to translation) with J.-L. Loday's associahedron [Lod04] and the zonotope Zono 1 (n) coincides with the parallelotope generated by the simple roots e i+1 − e i .

2.2
The geometry of the surjections ins k , can k , and rec k The main geometric connection between the three polytopes Perm(n), Brick k (n) and Zono k (n) is given by their normal fans. Remember that a polyhedral fan is a collection of polyhedral cones of R n closed under faces and which intersect pairwise along faces. The (outer) normal cone of a face F of a polytope P is the cone generated by the outer normal vectors of the facets of P containing F . Finally, the (outer) normal fan of P is the collection of the (outer) normal cones of all its faces.
The incidence cone C( ) and the braid cone C ♦ ( ) of an poset are the polyhedral cones defined by C( ) := cone {e i − e j | for all i j} and C ♦ ( ) := x ∈ 1 1 ⊥ x i ≤ x j for all i j . These two cones lie in the space 1 1 ⊥ := {x ∈ R n | x|1 1 = 0} and are polar to each other. For a permutation τ ∈ S n (resp. a twist T ∈ AT k (n), resp. an orientation O ∈ AO k (n)), we slightly abuse notation to write C(τ ) (resp. C(T), resp. C(O)) for the incidence cone of the chain τ 1 · · · τ n (resp. of the transitive closure of the contact graph T # , resp. of the transitive closure of O). We define similarly the braid cone C ♦ (τ ) (resp. C ♦ (T), resp. C ♦ (O)).
Using these normal fans, one can interpret geometrically the maps ins k , can k , and rec k as follows.
Proposition 27 The insertion map ins k : S n → AT k (n), the k-canopy can k : AT k (n) → AO k (n) and the k-recoil map rec k : S n → AO k (n) are characterized by Finally, the lattices studied in Section 1 also appear naturally in the geometry of these polytopes.

The Malvenuto-Reutenauer Hopf algebra FQSym on permutations
We briefly recall here the definition and some elementary properties of C. Malvenuto and C. Reutenauer's Hopf algebra on permutations [MR95], that we denote by FQSym. Let S := n∈N S n denote the set of all permutations, of arbitrary size.
For example,

Definition 29
We denote by FQSym the Hopf algebra with basis (F τ ) τ ∈S and whose product and coproduct are defined by Proposition 30 A product of weak order intervals in FQSym is a weak order interval: for any two weak order intervals [µ, ω] and [µ , ω ], we have µ≤τ ≤ω

Subalgebra of FQSym
We denote by Twist k the vector subspace of FQSym generated by the elements for all acyclic k-twists T. For example, for the (k, 5)-twists of Figure 3,  Theorem 31 Twist k is a Hopf subalgebra of FQSym, that we call k-twist Hopf algebra.

Example 32
• For k = 0, the bijection which sends the unique (0, n)-twist to X n /n! defines an isomorphism from the 0-twist algebra Twist 0 to the polynomial ring K[X].
We now aim at understanding the product and the coproduct in Twist k directly on k-twists. Although they are not always as satisfactory, our descriptions naturally extend classical results on the binary tree Hopf algebra PBT described in [LR98,AS06,HNT05].
Product. To describe the product in Twist k , we need the following notation. For a (k, n)-twist T and a (k, n )-twist T , we denote by T\T the (k, n + n )-twist obtained by inserting T in the first rows and columns of T and by T/T the (k, n+n )-twist obtained by inserting T in the last rows and columns of T.
Proposition 33 For any acyclic k-twists T ∈ AT k (n) and T ∈ AT k (n ), we have P T · P T = S P S , where S runs over the interval between T\T and T/T in the (k, n + n )-twist lattice. Coproduct. Our description of the coproduct in Twist k is unfortunately not as simple as the coproduct in PBT. We need the following definition. A cut in a k-twist S is a set γ of edges of the contact graph S # such that any path in S # from a leaf to the root contains precisely one edge of γ. We then denote by A # (S, γ) (resp. B # (S, γ)) the restriction of the contact graph S # to the nodes above (resp. below) γ. Moreover, A # (S, γ) is the contact graph of the k-twist A(S, γ) obtained from S by deleting all pipes below γ in S # . Nevertheless, note that B # (S, γ) is not a priori the contact graph of a k-twist.
Proposition 34 For any acyclic k-twist S ∈ AT k (m), we have P S = γ τ P ins k (τ ) ⊗ P A(S,γ) , where γ runs over all cuts of S and τ runs over a set of representatives of the k-twist congruence classes of the linear extensions of B # (S, γ). Matriochka algebras. Consider the k-recoil algebra Rec k defined in [NRT11] as the Hopf subalgebra of FQSym generated by the elements X O := rec k (τ )=O F τ , for all acyclic orientations O of the graph G k (n) for all n ∈ N. The commutative diagram of Proposition 24 ensures that X O = can k (T)=O P T , and thus that Rec k is a Hopf subalgebra of Twist k .
Remark 35 Further properties of Twist k can be found in [Pil15]. In particular, we describe combinatorially in terms of (k, n)-twists the product and coproduct operations on the dual Hopf algebra Twist k * of Twist k , we study some multiplicative bases of Twist k and their indecomposable elements, and we observe a connection to integer point transform in certain cones related to k-twists.

k-twistiform algebras
In this section, we extend the notion of dendriform algebras [Lod01, Chap. 5] to k-twistiform algebras. In a dendriform algebra, the product · is decomposed into two partial products ≺ and satisfying: x ≺ y · z = x ≺ y ≺ z, x y ≺ z = x y ≺ z, and x y z = x · y z.
In our context, we will still decompose the product of FQSym (and of Twist k ) into partial products, but we will use 2 k partial products satisfying 3 k relations. We need to fix some natural notations on words. We denote by |W | the length of a word W . For a subset P of positions in W and a subset L of letters of W , we denote by W P the subword of W consisting only of the letters at positions in P and by W L the subword of W consisting only of the letters which belong to L.
Definition 36 A k-twistiform algebra is a vector space Alg endowed with a collection B := {≺, ≺ , } k of 3 k bilinear operations which satisfy the following k3 k−1 + 3 k relations: Split relations For any b, b ∈ {≺, ≺ , } * with |b| + |b | = k − 1, the operation b≺ b ∈ B decomposes into the operations b≺b ∈ B and b b ∈ B: x b≺ b y = x b≺b y + x b b y for all x, y ∈ Alg.
Associativity relations For any W ∈ {x, y, z} k , the operations satisfy the associativity relation x b W y b W z = x b W y b W z for all x, y, z ∈ Alg.
We define a k-twistiform structure on FQSym as follows. For an operation b ∈ B and two permutations τ ∈ S n and τ ∈ S n , we define τ b τ to be the set of permutations σ ∈ τ¡ τ such that for all i ∈ [k], we have σ n+n +1−i ≤ n if b i = ≺ while σ n+n +1−i > n if b i = . Finally, we define the operations B on the Hopf algebra FQSym itself by Proposition 38 The Hopf algebra FQSym, endowed with the operations B described above, defines a k-twistiform algebra. The product of FQSym is then given by · = ≺ k . The subalgebra Twist k of FQSym is stable by the operations B and therefore inherits a k-twistiform structure.
This statement motivates the study of the k-twistiform operad. Several similar operads generalizing the dendriform and the diassociative operads are studied in [HP15].