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Associahedron Associahedron Jean-Louis Loday CNRS, Strasbourg April 23rd, 2005 Clay Mathematics Institute 2005 Colloquium Series Polytopes 7T 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 TTTz*vv 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 OOz'tt 3 Poset structure of Y4 on K cGG /gOO GG OOO GG OOO GG OOO GG Ot: GG tt GG tt GG tt GG tt ×c/G/G G tt /GG GG tt / GG GG tt // GG GG tt / GG G /tt / 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 7T 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.
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