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New Hopf structures on binary trees.

Introduction

In the past 30 years, there has been an explosion of interest in combinatorial Hopf algebras related to the classical ring of symmetric functions. This is due in part to their applications in combinatorics and representation theory, but also in part to a viewpoint expressed in the elegant commuting diagram

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Namely, much information about an object may be gained by studying how it interacts with its surroundings. From this picture, we focus on the right edge, [??]Sym : QSym. We factor this map through finer and finer structures (some well-known and some new) until this edge is replaced by a veritable zoo of Hopf structures. A surprising feature of our results is that each of these factorizations may be given geometric meaning--they correspond to successive polytope quotients from permutahedra to hypercubes.

The (known) cast of characters

Let us reacquaint ourselves with some of the characters who have already appeared on stage.

[??]Sym - the Hopf algebra introduced by Malvenuto and Reutenauer [13] to explain the isomorphism QSym [equivalent] [(NSym).sup.*]. A graded, noncommutative, noncocommutative, self-dual Hopf algebra, with basis indexed by permutations, it offers a natural setting to practice noncommutative character theory [4].

YSym--the (dual of the) Hopf algebra of trees introduced by Loday and Ronco [11]. A graded, noncommutative, noncocommutative Hopf algebra with basis indexed by planar binary trees, it is important for its connections to the Connes-Kreimer renormalization procedure.

QSym--The Hopf algebra of quasisymmetric functions introduced by Gessel [9] in his study of P-partitions. A graded, commutative, noncocommutative Hopf algebra with basis indexed by compositions, it holds a special place in the world of combinatorial Hopf algebras [1].

The new players

In this extended abstract, we study in detail a family of planar binary trees that we call bi-leveled trees, which possess two types of internal nodes (circled or not, subject to certain rules). These objects are the vertices of Stasheff's multiplihedra [18], originating from his study of [A.sub.[infinity]] categories. The multiplihedra were given the structure of CW-complexes by Iwase and Mimura [10] and realized as polytopes later [8]. They persist as important objects of study, among other reasons, because they catalog all possible ways to multiply objects in the domain and range of a function f, when both have nonassociative multiplication rules. More recently, they have appeared as moduli spaces of "stable quilted discs" [14].

In Section 2, we define a vector space MSym with basis indexed by these bi-leveled trees. We give MSym a module structure for SSym by virtue of the factorization

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(evident on the level of planar binary trees) and a splitting MSym [??] SSym. We also show that MSym is a Hopf module for YSym and we give an explicit realization of the fundamental theorem of Hopf modules. That is, we find the coinvariants for this action. Our proof, sketched in Section 3, rests on a result about poset maps of independent interest.

We conclude in Section 4 with a massive commuting diagram--containing several new families of planar binary trees--that further factors the map from SSym to QSym. The remarkable feature of this diagram is that it comes from polytopes (some of them even new) and successive polytope quotients. Careful study of the interplay between the algebra and geometry will be carried out in future work.

1 Basic combinatorial data

1.1 Ordered and planar binary trees

We recall a map [tau] from permutations [??]. = [[union].sub.n] [[??].sub.n] to planar binary trees y. = [[union].sub.n][y.sub.n] that has proven useful in many contexts [19, 12]. Its behavior is best described in the reverse direction as follows. Fix a tree t [member of] [Y.sub.n]. The n internal nodes of t are equipped with a partial order, viewing the root node as maximal. An ordered tree is a planar binary tree, together with a linear extension of the poset of its nodes. These are in bijection with permutations, as the nodes are naturally indexed left-to-right by the numbers 1, ..., n. The map [tau] takes an ordered tree (permutation) to the unique tree whose partial order it extends.

Example 1 The permutations 1423, 2413, and 3412 share a common image under [tau]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

There are two right inverses to t that will be useful later. Let min(t) (respectively, max(t)) denote the unique 231-avoiding (132-avoiding) permutation mapping to t under [tau]. Loday and Ronco show that [[tau].sup.-1](t) is the interval [min(t), max(t)] in the weak Bruhat order on the symmetric group [12, Thm. 2.5], and that min and max are both order-preserving with respect to the Tamari order on [Y.sub.n].

1.2 Bi-leveled trees and the multiplihedra

We next describe a family of bi-leveled trees intermediate between the ordered and unordered ones. These trees arrange themselves as vertices of the multiplihedra M. = [[union].sub.n] [M.sub.n], a family of polytopes introduced by Stasheff in 1970 [18] (though only proven to be polytopes much later [8]). Stasheff introduced this family to represent the fundamental structure of a weak map [florin] between weak structures, such as weak n-categories or [A.sub.n] spaces. The vertices of [M.sub.n] correspond to associations of n objects, pre- and post- application of [florin], e.g., ([florin] (a) [florin] (b)) [florin] (c) and [florin] (a) [florin] (bc). This leads to a natural description of [M.sub.n] in terms of "painted binary trees" [5], but we use here the description of Saneblidze and Umble [16].

A bi-leveled tree is a pair (t, C) with t [member of] [Y.sub.n] and C [bar.[subset]] [n] designating some nodes of t as lower than the others (indexing the nodes from left-to-right by 1, ..., n). Viewing t as a poset with root node maximal, C is an increasing order ideal in t where the leftmost node is a minimal element. Graphically, C indexes a collection of nodes of t circled according to the rules: (i) the leftmost node is circled and has no circled children; (ii) if a node is circled, then its parent node is circled.

Define a map [beta] from [[??].sub.n] to [M.sub.n] as follows. Given a permutation [sigma] = [[sigma].sub.1] [[sigma].sub.2] ... [[sigma].sub.n], first represent [sigma] as an ordered tree. Next, forget the ordering on the nodes, save for circling all nodes [[sigma].sub.i] with [[sigma].sub.i] > [[sigma].sub.1].

Example 2 Consider again the permutations 1423, 2413, and 3412 of Example 1. Viewed as ordered trees, their images under [beta] are distinct:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Denote by [phi] the map from bi-leveled trees to trees that forgets which nodes are circled. The map [phi] helps define a partial order on bi-leveled trees that extends the Tamari lattice on planar binary trees: say that the bi-leveled tree s precedes the bi-leveled tree t in the partial order if [phi](s) [less than or equal to] [phi](t) and the circled nodes satisfy [C.sub.t] [bar.[subset]] [C.sub.s]. We call this the weak order on bi-leveled trees. See Figure 1 below for an example.

The equality [phi] o [beta] = t is evident. Remarkably, this factorization

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 1 OMITTED]

extends to the level of face maps between polytopes (see Figure 2). Our point of departure was the observation that it is also a factorization as poset maps.

[FIGURE 2 OMITTED]

1.3 Dimension enumeration

Fix a field k of characteristic zero and let [??]Sym = [[direct]n[greater than or equal to]0] [??]Symn denote the graded vector space whose nth graded piece has the "fundamental" basis {[F.sub.[sigma]] | [sigma] an ordered tree in [[??].sub.n]}. Define MSym and ySym similarly, replacing [[??].sub.n] by [M.sub.n] and [Y.sub.n], respectively. We follow convention and say that [[??]Sym.sub.0] and [ySym.sub.0] are 1-dimensional. By contrast, we agree that [MSym.sub.0] = {0}. (See [7] for categorical rationale; briefly, Stasheff's [M.sub.1] is already 0-dimensional, so [M.sub.0] has no clear significance.)

In Section 2, we give these three vector spaces a variety of algebraic structures. Here we record some information about the dimensions of the graded pieces for later reference.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Of course, [C.sub.n] is the nth Catalan number. The enumeration of bi-leveled trees is less familiar: the nth term satisfies [A.sub.n] = [C.sub.n-1] + [[summation].sup.n-1.sub.k=1] [A.sub.i] [A.sub.n-i] [17, A121988]. A little generating function arithmetic can show that the quotient of (2) by (3) expands as a power series with nonnegative coefficients,

[Hilb.sub.q](MSym)/[Hilb.sub.q](ySym) = q + [q.sup.2] + 3[q.sup.3] + 11[q.sup.4] + 44[q.sup.5] + ....

We will recover this with a little algebra in Section 2.3. The positivity of the quotient of (1) by (3) is established by [3, Theorem 7.2].

2 The Hopf module MSym

Let [tau], [beta], and [phi] be the maps between the vector spaces [??]Sym, MSym, and YSym induced by [tau], [beta], and [phi] on the fundamental bases. That is, for permutations [sigma] and bi-leveled trees t, we take

[tau]([F.sub.[sigma]]) = [[F.sub.[tau]([sigma])] [beta]([[F.sub.[sigma]])= [F.sub.[beta]([sigma])] [phi]([F.sub.t]) = [F.sub.[phi](t)]

Below, we recall the product and coproduct structures on the Hopf algebras [??]Sym and ySym. In [13] and [11], these were defined in terms of the fundamental bases. Departing from these definitions, rich structural information was deduced about [??]Sym, ySym, and the Hopf algebra map [tau] between them in [2, 3]. This information was revealed via a change of basis--from fundamental to "monomial"--using Mobius inversion. We take the same tack below with MSym and meet with similar success.

2.1 The Hopf algebras [??]Sym and ySym

Following [3], we define the product and coproduct structures on [??]Sym and ySym in terms of p-splittings and graftings of trees. A p-splitting of a tree t with n nodes is a forest (sequence) of p + 1 trees with n nodes in total. This sequence is obtained by choosing p leaves of t and splitting them (and all parent branchings) right down to the root. By way of example, consider the 3-splitting below (where the third leaf is chosen twice and the fifth leaf is chosen once).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Denote a p-splitting of t by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The grafting of a forest ([t.sub.0], [t.sub.1]; ..., [t.sub.p]) onto a tree with p nodes is also best described in pictures; for the forest above and s = t(213), the tree

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is the grafting of ([t.sub.0], [t.sub.1]; ..., [t.sub.p]) onto s, denoted ([t.sub.0], [t.sub.1]; [t.sub.2], [t.sub.3])/s. Splittings and graftings of ordered trees are similarly defined. One remembers the labels originally assigned to the nodes of t in a p-splitting, and if t has q nodes, then one increments the labels of s by q in a grafting ([t.sub.0], [t.sub.1]; [t.sub.2], [t.sub.3])/s. See [3] for details.

Definition 3 Fix two ordered or ordinary trees s and t with p and q internal nodes, respectively. We define the product and coproduct by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(In the coproduct for ordered trees, the labels in t0 and t1 are reduced to be permutations of [absolute value of [t.sub.0] and [absolute value of [t.sub.0].)

2.2 Module and comodule structures

We next modify the structure maps in (5) to give MSym the structure of (left) [??]Sym-module and (right) YSym-Hopf module. Given a bi-leveled tree b, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represent any p- splitting of the underlying tree, together with a circling of all nodes in each [b.sub.i] that were originally circled in b.

Definition 4 (action of [??]Sym on MSym) For w [member of] [[??].sub..] and s [member of] [M.sub.p], write b = [beta](w) and set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the circling rules in ([b.sub.0], [b.sub.1], ..., [b.sub.p])/s are as follows: every node originating in s is circled whenever [absolute value of [b.sub.0] > 0, otherwise, every node originating in b = [beta](w) is uncircled.

This action may be combined with any section of [beta] to define a product on MSym. For example,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 5 The action [??]Sym [cross product] MSym [right arrow] MSym and the product MSym [cross product] MSym [cross product] MSym are associative. Moreover, putting [MSym.sub.0] := k, they make [beta] into an algebra map that factors [tau].

Unfortunately, no natural coalgebra structure exists on MSym that makes [beta] into a Hopf algebra map.

Definition 6 (action and coaction of ySym on MSym) Given b [member of] M., let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denote a p-splitting satisfying [absolute value of [b.sub.0] > 0. For s [member of] [y.sub.p], set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where in ([b.sub.0], [b.sub.1], ..., [b.sub.p])/s every node originating in s is circled, and in [phi]([b.sub.1]) all circles are forgotten. Example 7 In the fundamental bases of MSym and ySym, the action looks like

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

while the coaction looks like

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The significance of our definition of [rho] will be seen in Corollary 11. Our next result requires only slight modifications to the original proof that Y Sym is a Hopf algebra (due to the restricted p-splittings).

Theorem 8 The maps * : MSym [cross product] YSym [right arrow] MSym and [rho] : MSym [right arrow] MSym [cross product] YSym are associative and coassociative, respectively. They give MSym the structure of YSym-Hopf module. That is, [rho]([F.sub.b] * [F.sub.s]) = [rho]([F.sub.b]) * [DELTA]([F.sub.s]).

2.3 Main results

We next introduce "monomial bases" for [??]Sym, MSym, and YSym. Given t [member of] [M.sub.n], define

[M.sub.t] = [summation over t[less than or equal to]]t'] [mu](t, t')[F.sub.t'], t<t'

where [mu](*, *) is the Mobius function on the poset [M.sub.n]. Define the monomial bases of [??]Sym and YSym similarly (see (13) and (17) in [3]). The coaction [rho] in this basis is particularly nice, but we need a bit more notation to describe it. Given t [member of] [M.sub.p] and s [member of] [Y.sub.q], let t\s denote the bi-leveled tree on p + q internal nodes formed by grafting the root of s onto the rightmost leaf of t.

Theorem 9 Given a bi-leveled tree t, the coaction [rho] on [M.sub.t] is given by [rho]([M.sub.t]) = [summation over t=t'\s] [M.sub.t'] [cross product] [M.sub.s].

Example 10 Revisiting the trees in the previous example, the coaction in the monomial bases looks like

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Recall that the coinvariants of a Hopf module M over a Hopf algebra H are defined by [M.sup.co] = {m [member of] M | [rho](m) = m [cross product] 1}. The fundamental theorem of Hopf modules provides that M [equivalent] [M.sup.co] [cross product] H. The monomial basis of MSym demonstrates this isomorphism explicitly.

Corollary 11 A basis for the coinvariants in the Hopf module MSym is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where T comprises the bi-leveled trees with no uncircled nodes on their right branches.

This result explains the phenomenon observed in (4). It also parallels Corollary 5.3 of [3] to an astonishing degree. There, the right-grafting idea above is defined for pairs of planar binary trees and used to describe the coproduct structure of ySym in its monomial basis.

3 Towards a proof of the main result

We follow the proof of [3, Theorem 5.1], which uses properties of the monomial basis of 6Sym developed in [2] to do the heavy lifting. In [3], the section [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of [tau] is shown to satisfy [tau]([M.sub.max(t)]) = [M.sub.t] and [tau] ([M.sub.[sigma]]) = 0 if [sigma] is not 132-avoiding. This was proven using the following result about Galois connections.

Theorem 12 ([15, Thm. 1]) Suppose P and Q are two posets related by a Galois connection, i.e., a pair of order-preserving maps [phi] : P [right arrow] Q and [gamma] : Q [right arrow] P such that for any v [member of] P and t [member of] Q, [phi](v) [less than or equal to] t [??] v [less than or equal to] [gamma](t). Then the Mobius functions [mu]P and [mu]Q are related by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

There is a twist in our present situation. Specifically, no Galois connection exists between [[??].sub.n] and [M.sub.n]. On the other hand, we find that no order-preserving map [iota] : M. [??] [??]. satisfies [beta]([M.sub.[iota](t)]) = [M.sub.t]. Rather,

[beta]([summation over [sigma][member of][[beta].sup.1](t)] [M.sub.[sigma]]) = [M.sub.t]. (8)

This fact is the key ingredient in our proof of Theorem 9. Its verification required modification of the notion of Galois connection--a relationship between posets that we call an interval retract (Section 3.2).

3.1 Sections of the map [beta] : [??]. [right arrow] M.

Bi-leveled trees t are in bijection with pairs {s, s}, where s is a planar binary tree, with p nodes say, and s = ([s.sub.1], ..., [s.sub.p]) is a forest (sequence) of planar binary trees. In the bijection, s comprises the circled nodes of t and [s.sub.i] is the binary tree (of uncircled nodes) sitting above the ith leaf of s. For example,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A natural choice for a section [iota] : [M.sub.n] [right arrow] [[??].sub.n] would be to, say, build min([s.sub.i]) and min(s4) for each i and splice these permutations together in some way to build a word on the letters {1, 2, ... n}. Let mm(t) denote the choice giving s1 smaller letters than [s.sub.2], [s.sub.2] smaller letters than [s.sub.3], ..., [s.sub.p-1] smaller letters than [s.sub.p], and [s.sub.p] smaller letters than s:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This choice does not induce a poset map. The similarly defined MM also fails (chosing maximal permutations representing s and s), but Mm has the properties we need:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We define this map carefully. Given t [member of] [y.sub.n] and any subset S [bar.[subset]] N of cardinality n, write minS(t) for the image of min(t) under the unique order-preserving map from [n] to S; define maxS(t) similarly.

Definition 13 [The section Mm] Let t [left and right arrow] {s, s} be a bi-leveled tree on n nodes with p circled nodes. Write u = [u.sub.1] ... [u.sub.p] = [min.sub.[a, b]] (s) for [a, b] = {n - p + 1, ..., n} and write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the intervals [[a.sub.i]; [b.sub.i]] are defined recursively as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, define Mm(t) by the concatenation Mm(t) = [u.sub.1][v.sup.1][u.sub.2][v.sup.2] ... [u.sup.p][v.sup.p].

Remark 14 Alternatively, Mm(t) is the unique w [member of] [[beta].sup.-1](t) avoiding the pinned patterns [bar.0]231, [bar.3]021, and [bar.2]031, where the underlined letter is the first letter in w. The first two patterns fix the embeddings of [S.sub.i] (0 [less than or equal to] i [less than or equal to] p), the last one makes the letters in [s.sub.i] larger than those in [s.sub.i+1] (1 [less than or equal to] i < p).

The important properties of Mm(t) are as follows.

Proposition 15 The section [iota] : [M.sub.n] [right arrow] [[??].sub.n] given by [iota](t) = Mm(t) is an embedding of posets. The map [beta] : [[??].sub.n] [right arrow] [M.sub.n] satisfies [beta]([iota](t)) = t for all t [member of] [M.sub.n] and [[beta].sup.-1](t) [bar.[subset]] [[??].sub.n] is the interval [mm(t), MM(t)].

3.2 Interval retracts

Let [phi] : P [right arrow] Q and [gamma] : Q [right arrow] P be two order-preserving maps between a lattice P and a poset Q. If

[for all]t [member of] Q [phi]([gamma](t)) = t and [[phi].sup.p-1](t) is an interval,

then we say that [phi] and [gamma] demonstrate P as an interval retract of Q.

Theorem 16 f P and Q are two posets related by an interval retract ([phi], [gamma]), then the Mobius functions [mu]P and [mu]Q are related by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The proof of Theorem 16 exploits Hall's formula for Mobius functions. An immediate consequence is a version of (8) for any P and Q related by an interval retract. Verifying that ([beta], Mm) is an interval retract between [[??].sub.n] and [M.sub.n] (Proposition 15) amounts to basic combinatorics of the weak order on [[??].sub.n].

4 More families of binary trees and their polytopes

We have so far ignored the algebra QSym of quasisymmetric functions advertised in the introduction. A basis for its nth graded piece is naturally indexed by compositions of n, but may also be indexed by trees as follows. To a composition ([a.sub.1], [a.sub.2], ...), say (3, 2, 1, 4), we associate a sequence of left-combs

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

i.e., trees with [a.sub.i] leaves and all internal leaves rooted to the rightmost branch and left-pointing. These may be hung on another tree, a right-comb with right-pointing leaves, to establish a bijection between compositions of n and "combs of combs" with n internal nodes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To see how QSym and the hypercubes fit into the picture, we briefly revisit the map [beta] of Section 1.2.

We identified bi-leveled trees with pairs {s, s}, where s is the tree of circled nodes and s = ([s.sub.0], ..., [s.sub.p]) is a forest of trees (the uncircled nodes). Under this identification, [beta] may be viewed as a pair of maps ([tau], [tau])--with the first factor [tau] making a (planar binary) trees out of the nodes greater than or equal to p[sigma].sub.1], and the second factor [tau] making trees out of the smaller nodes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

See also Figure 3. Two more fundamental maps are [[gamma].sub.r] and [[gamma].sub.l], taking trees to (right- or left-) combs, e.g.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Figure 3 displays several combinations of the maps [tau], [[gamma].sub.r], and [[gamma].sub.l]. The algebra QSym corresponds to the terminal object there--the set denoted { ... combs .../comb}.

The new binary tree-like structures appearing in the factorization of [??]Sym [??] QSym (i.e., those trees not appearing on the central, vertical axis of Figure 3) will be studied in upcoming papers. It is no surprise that [??]Sym [??] QSym factors through so many intermediate structures. What is remarkable, and what our binary tree point-of-view reveals, is that each family of trees in Figure 3 can be arranged into a family of polytopes. See Figure 4. The (Hopf) algebraic and geometric implications of this phenomenon will also be addressed in future work.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

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[19] Andy Tonks, Relating the associahedron and the permutohedron, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math., vol. 202, Amer. Math. Soc., Providence, RI, 1997, pp. 33-36.

Stefan Forcey, (1) Aaron Lauve (2) and Frank Sottile (2) ([dagger])

(1) Department of Mathematics Tennessee State University 3500 John A Merritt Blvd Nashville, Tennessee 37209

(2) Department of Mathematics Texas A&M University MS 3368 College Station, Texas 77843

([dagger]) Sottile supported by NSF grant DMS-0701050

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