Associahedron

Jean-Louis Loday

CNRS, Strasbourg

April 23rd, 2005

Clay Mathematics Institute

2005 Colloquium Series Polytopes

T7 4 × 7TTTT 4 ×× 77 TT 44 × 7 simplex 4 ×× 77 44 × 7 ··· 4 ×× 77 ×× 7

o o ooo ooo cube o o ··· oo ooo

oo?? ooo ?? ? ?? ? ??  OOO  OO

ooooOOO oooooo OO?  ?? oOO  ooo OOO  o  OOO oo?? permutohedron  OOooo ? ···    OO o ? O O   OOO ooo ?? OOO   o OOO ?? oo OO ?ooo

n = 2 3 ···

Permutohedron := convex hull of (n+1)! points

n+1 (σ(1), . . . , σ(n + 1)) ∈ R Parenthesizing

X= topological space with product (a, b) 7→ ab

Not associative but associative up to homo- topy

(ab)c • ) • a(bc)

With four elements:

((ab)c)d H jjj HH jjjj HH jjjj HH tjj HH (a(bc))d HH HH HH HH H$ vv (ab)(cd) vv vv vv vv (( ) ) TTT vv a bc d TTTT vv TTT vv TTTzv* v a(b(cd))

We suppose that there is a homotopy between the two composite paths, and so on. Jim Stasheff

Staheff’s result (1963): There exists a cellular complex such that – vertices in bijection with the parenthesizings – edges in bijection with the homotopies – 2-cells in bijection with homotopies of com- posite homotopies – etc, and which is homeomorphic to a ball.

Problem: construct explicitely the Stasheff com- plex in any dimension.

oo?? ooo ?? ?? ??  • OOO  OO

n = 0 1 2 3 Planar binary trees

(see R. Stanley’s notes p. 189)

Planar binary trees with n + 1 leaves, that is n internal vertices:   n o ? ?? ? ?? ??  ??  Y0 = { | } ,Y1 =  ,Y2 = ? , ? ,

( ) ? ? ? ? ? ? ? ? ? ? ??   ??   ??  ?? ??  ?? ??  ??  ??  ??  ?? ? ?? ? Y3 = ? , ? , ? , ? , ? .

Bijection between planar binary trees and paren- thesizings:

x0 x1 x2 x3 x4 RRR ll ll RRR ll RRlRll lll RRlll RRR lll lll lRRR lll RRR lll RRlll

(((x0x1)x2)(x3x4))

The notion of grafting s t 3 33 t ∨ s = 3 Associahedron

n To t ∈ Yn we associate M(t) ∈ R : n M(t) := (a1b1, ··· , aibi, ··· , anbn) ∈ R ai = # leaves on the left side of the ith vertex bi = # leaves on the right side of the ith vertex

Examples:

 ?   ? ??  ?? ??  ??  M(  ) = (1),M ? = (1, 2),M ? = (2, 1),

 ?   ? ?  ??   ??  ??  ??  M ? = (1, 2, 3),M ? = (1, 4, 1).

For the tree corresponding to (((x0x1)x2)(x3x4)): (1 × 1, 2 × 1, 3 × 2, 1 × 1) = (1, 2, 6, 1)

Definition of the associahedron: n−1 K := convex hull of M(t), t ∈ Yn Stasheff polytope

Theorem The associahedron is isomorphic to the Stasheff complex as a cellular complex.

GG OOO J  GG t oo JJ /G/G GG ttt ooo JJ / GG t  JJ / G    J // G/   tt ? //   • OO ttt ??   OOO tt ??  t ? 

n = 0 1 2 3

3 K GG OO  GG OOO  GG OOO  GG OOO  GG Ot  GG tt  GG tt   GG tt   GG tt  /G/G G tt  /GG GG tt  / GG GG tt  // GG GG tt  / GG G tt  / GG    // GG    / GG    // GG    / GG    / GG    // GG    / GG   // //   / /   // //   / /   ?? //   ?? /   ?? //   ?? /   ??   ??  ??  ??  ??  ??  ??  ?  K3 Construction of Kn+1 out of Kn

- Start with Kn, boundary = cellular sphere - cells of the boundary of the form Kp × Kq where p + q = n − 1 - enlargement of Kp × Kq, make it Kp × Kq+1 - take the cone over the resulting space - check that this is Kn+1.

Example n = 1:

1 – K oo ooo

– K1 enlarged ooJJJ ooo JJ JJJ

– Cone over K1 enlarged = K2 ooJJJ ooo JJ JJJ Example n = 2:

– K2   //  / 

2 – enlarged GG OO K  G O /G GG ttt /GG G tt  // GG    / G/   /? /   ?? /   ??  ? 

2 3 – Cone over enlarged = GG OO K K  G O /G GG ttt /GG G tt  // GG    / G/   /? /   ?? /   ??  ? 

Exercise: # of simplices in Kn is (n + 1)n−1. Associahedron and permutohedron

Y˜n = set of p.b. leveled trees with n + 1 leaves

??  //  vs //  ??  ??  /  /  ??  ?? //  //  ? ?? //  //   ?? /  /   ??  ??  ??  ??  ? ?

∼ φ : Sn = Y˜n −→ Yn (forget the levels)

Proposition Let C = center of Pn−1 n+1 n+1 C = ( 2 ,..., 2 ). Then on has −−−−→ −−−−−→ CM(t) = X CM(σ) . σ∈φ−1(t)

J oo JJ oo JJ ooo JJ oo ? JJ  JJ  JJ  JJ ? t/ ?? tt ?? tt ? tt OO tt OOO tt OO tt OOtt Inversion of power series

2 3 n+1 f(x) = x + a1x + a2x + ··· + anx + ··· 2 3 n+1 g(x) = x + b1x + b2x + ··· + bnx + ··· such that f(g(x)) = x bn = polynomial in the coefficients ai, 1 ≤ i ≤ n

b1 = −a1 2 b2 = 2a1 − a2 3 b3 = −5a1 + 5a1a2 − a3 4 2 2 b4 = 14a1 − 21a1a2 + 6a1a3 + 3a2 − a4 ··· = ··· P X ni n1 nk bn = (−1) λ(n1, . . . , nk)a1 ··· ak where n1 + 2n2 + ··· + knk = n

n−1 Claim: λ(n1, . . . , nk) = # cells in K iso- morphic to (K0)n1 × · · · × (Kk−1)nk

Examples: λ(0,..., 0, 1) = 1 1 2n λ(n) = Catalan number Cn = n+1 n Poset structure

Partial order on the set Yn of p.b. trees

? ? ?? ??  ??  In Y2: ? −→ ?

In Yn: change, locally in the tree t,

? ? ?? ??  ??  ? into ? to get s covering relation: t → s

Examples:

J oo JJ ooo JJ oo JJ ooo JJ woo JJ JJ JJ JJ JJ t$ tt tt tt tt OOO tt OO tt OOO tt OO tt OOzt' t 3 Poset structure of Y4 on K

cGG gO/ O  GG OOO  GG OOO  GG OOO  GG Ot:  GG tt  GG tt   GG tt   GG tt  /Gc×/G G tt  /GG GG tt  / GG GG tt  // GG GG tt  / GG G t/ t  / GG    // GG    / GG    // GG    / GG    / GG    // GG    / GG   // /×/   / /   // //   / /   _? ? // g/×   ?? /   ?? //   ?? /   ??  ×/ ×? ??  ??  ??  ??  ??  ??  ? /  Algebraic structure

K[Yn] = vector space over K spanned by p.b.trees having n vertices

L Define inductively an operation on n≥0 K[Yn], t ∗ s := tl ∨ (tr ∗ s) + (t ∗ sl) ∨ sr, | = 1

Example:

? ?? ? ?? ?? ?? ?? ?? ?? ??  ??   ∗  = |∨(|∗ )+(  ∗|)∨| = |∨ +  ∨| = ? + ?

Prop The operation ∗ is associative and unital

Theorem

t ∗ s = X x t/s≤x≤t\s t/s “over” operation, t\s “under” operation Dendriform algebras

Define t ≺ s := tl∨(tr∗s) and t s := (t∗sl)∨sr, so t ∗ s = t ≺ s + t s

Prop The operations ≺ and satisfy the fol- lowing relations  (x ≺ y) ≺ z = x ≺ (y ∗ z),  (x y) ≺ z = x (y ≺ z),  (x ∗ y) z = x (y z).

Definition A dendriform algebra is a vector space A over K equipped with two operations ≺ and satisfying the three relations above.

Theorem The dendriform algebra L ( n≥0 K[Yn], ≺, ) is the free dendriform alge- ? bra on one generator, namely the tree ? .

? Hint: t ∨ s = t ? ≺ s Applications of dendriform algebras

The dendriform algebras are involved in many topics:

- shuffles and noncommutative shuffles,

- preLie and brace algebras (algebraic topol- ogy),

- Hopf algebras, noncommutative version of Connes and Kreimer (theoretical physics),

- combinatorics (nc symmetric functions)

- arithmetic of trees (arithmetree)

- series indexed by trees (differential equations) Dendriform and preLie

Definition preLie algebra: (A, ◦) such that (x ◦ y) ◦ z − x ◦ (y ◦ z) = (x ◦ z) ◦ y − x ◦ (z ◦ y)

Claim 1: [x, y] := x ◦ y − y ◦ x is a Lie bracket

Claim 2: x ◦ y := x ≺ y − y x is a preLie product

Dend / preLie

  As / Lie

Proof. x ≺ y + x y x ∗ y − y ∗ x = = x ◦ y − y ◦ x

−y x − y ≺ x Series indexed by trees

Power series: 2 n f(x) = a1x + a2x ··· + anx + ··· , n ∈ N

Dendriform series: t f(x) = a1x + ··· + atx + ··· , t ∈ Y∞

• Addition: OK (term by term),

• Multiplication: xtxs = xt∗s

• Composition: f(g(x)) =? consequence of the Theorem about freeness: what is g(x)t for a p.b. tree t ? Write t as ? (generalized) product of the generator tree ? , ? then replace ? by g(x) and compute. Families of polytopes

•

4 T7 4 × 7TTTT 44 ×× 77 TT 44 ×× 77 44 ×× 77 ×× 77 • ×× 7

oo oo ooo ooo

• oo ooo

GG OOO J  GG t oo JJ /G/G GG ttt ooo JJ / GG t  JJ / G    J // G/   tt ? //   • OO ttt ??   OOO tt ??  t ? 

o oOO oooooo OOO ooOOO o o ?? ooo OO  ?   OOO oo?? • O  OOooo ? OOO ooo    Ooo ? OO   ?? OOO   OOO ?? oo OO ?ooo n = 0 1 2 3 End

Many thanks for your attention !

GG OO  GG OOO  GG OOO  GG OOO  GG Ot  GG tt  GG tt   GG tt   GG tt  /G/G G tt  /GG GG tt  / GG GG tt  // GG GG tt  / GG G tt  / GG    // GG    / GG    // GG    / GG    / GG    // GG    / GG   // //   / /   // //   / /   ?? //   ?? /   ?? //   ?? /   ??   ??  ??  ??  ??  ??  ??  ?  http://www-irma.u-strasbg.fr/ loday/ Associahedron

Jean-Louis Loday

Polytopes Parenthesizing Jim Stasheff Planar binary trees Associahedron Stasheff polytope K3 Construction of Kn+1 out of Kn Associahedron and permutohedron Inversion of integral series Poset structure 3 Poset structure of Y4 on K Algebraic structure Dendriform algebras Applications of dendriform algebras Dendriform and preLie Series indexed by trees Families of polytopes