An operational calculus for the Mould operad
Frédéric Chapoton, Florent Hivert, Jean-Christophe Novelli, Jean-Yves Thibon
aa r X i v : . [ m a t h . QA ] O c t AN OPERATIONAL CALCULUS FOR THE Mould OPERAD
FR´ED´ERIC CHAPOTON, FLORENT HIVERT,JEAN-CHRISTOPHE NOVELLI,AND JEAN-YVES THIBON
Abstract.
The operad of moulds is realized in terms of an operational calculusof formal integrals (continuous formal power series). This leads to many simpli-fications and to the discovery of various suboperads. In particular, we prove aconjecture of the first author about the inverse image of non-crossing trees in thedendriform operad. Finally, we explain a connection with the formalism of non-commutative symmetric functions. Introduction A mould , as defined by Ecalle, is a “function of a variable number of variables”,that is, a sequence f n ( u , . . . , u n ) of functions of n (continuous or discrete) variables.He developed around this notion a versatile formalism which is an essential technicaltool in his theory of resurgence [10, 11] and in his later work on polyzetas [12, 13, 14](see the lecture notes [7] for an elementary introduction).In [6], the first author constructed an operad Mould from the set of (rational)moulds, and identified several (old and new) suboperads of it.The aim of this article is to introduce an operational calculus on formal integrals,which allows to simplify considerably the arguments of [6], and also to obtain furtherresults. In particular, we find that the operad Zinbiel introduced by Loday [20] is asub-operad of
Mould . As Zinbiel is based on permutations, this allows to considerthe elements of the algebra
FQSym of free quasi-symmetric functions as moulds.For instance, the classical Lie idempotents of Dynkin, Solomon and Klyachko giveinteresting examples of alternal moulds. We also find some other new suboperads,prove conjecture 5.7 of [6], and obtain some new examples of moulds.This article is a continuation of [6]. In a few examples, we shall assume that thereader is familar with the notation of [18, 8].2.
Moulds as nonlinear operators
Let H be a vector space of formal integrals(1) h ( t ) = Z h u t u − dµ ( u ) , where h u are homogeneous elements of degree u in some graded associative algebra A . We will only need the cases where µ is the Lebesgue measure on R or R + , orthe discrete measure on N , which gives back power series in t with noncommutative Key words and phrases.
Operads, Moulds, Trees, Noncommutative symmetric functions. coefficients. In these cases, the object (1) can be interpreted as a linear map
V → A on the vector space with basis ( t u ) for u in the support of µ , and the theory iscompletely similar to that of formal power series.A mould f = ( f n ( u , . . . , u n )) can be interpreted as a nonlinear operator F on H ,by setting(2) F [ h ] = X n ≥ Z · · · Z f n ( u , . . . , u n ) h u · · · h u n t u + ··· + u n dµ ( u ) · · · dµ ( u n ) . It will be convenient to set(3) H ( t ) = Z t h ( τ ) dτ = Z h u t u u dµ ( u ) . We also define the polarization of F as the collection of multilinear operators (the h ( i ) are arbitrary elements of H )(4) F n [ h (1) , . . . , h ( n ) ] = Z · · · Z f n ( u , . . . , u n ) h (1) u · · · h ( n ) u n t u + ··· + u n dµ ( u ) · · · dµ ( u n ) . Examples of moulds
In this section, we translate all the examples of [6] into the new formalism, andprovide some new ones.
Example 3.1.
The mould(5) f n ( u , . . . , u n ) = 1 u · · · u n corresponds to the operators(6) F n [ h (1) , . . . , h ( n ) ] = H (1) ( t ) · · · H ( n ) ( t ) . Example 3.2.
The time-ordered exponential(7) U ( t ) = T exp (cid:26)Z t h ( τ ) dτ (cid:27) , i.e. , the unique solution of U ′ ( t ) = U ( t ) h ( t ) with U (0) = 1, is given by the mould(8) f n ( u , . . . , u n ) = 1 u ( u + u ) · · · ( u + u + · · · + u n ) . Example 3.3.
More generally, for a permutation σ ∈ S n , the mould(9) f σ ( u , . . . , u n ) = 1 u σ (1) ( u σ (1) + u σ (2) ) · · · ( u σ (1) + u σ (2) + · · · + u σ ( n ) )integrates over the simplex ∆ σ ( t ) = { < t σ (1) < t σ (2) < · · · < t σ ( n ) < t } :(10) F σ [ h (1) , . . . , h ( n ) ] = Z ∆ σ ( t ) h (1) ( t ) · · · h ( n ) ( t n ) dt · · · dt n . It follows from the well-known decomposition of a product of simplices as a unionof simplices that these moulds form a subalgebra, isomorphic to the algebra of freequasi-symmetric functions
FQSym , under the correspondence f σ F σ (cf. [8]). N OPERATIONAL CALCULUS FOR THE
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Example 3.4.
To each planar binary tree T , we can associate an operator F T definedby F • [ h ] = H and, if T = T ∧ T has T and T as left and right subtrees(11) F T [ h ] = Z t F T [ h ]( τ ) h ( τ ) F T [ h ]( τ ) dτ . The kernels of these operators are the moulds associated to trees in [6], which canbe computed graphically as follows. All the leaves of T are labelled by 1, and theinternal nodes are labelled by t u i − , in such a way that flattening the tree yields the t u i − in their natural order(12) t u − VVVVVVVVVkkkkk t u − FFFxxx t u − qqq 444444 t u − pppp NNNN f T ( u , . . . , u n ) is obtained by evaluating the tree according to the followingrule: the outgoing flow of each node is the integral R t L ( τ ) v ( τ ) R ( τ ) dτ , where L ( t )and R ( t ) are the outputs of its left and right subtrees, and v ( t ) its label. For example,the above tree evaluates to(13) t u + u + u + u u u ( u + u )( u + u + u + u ) , as can be seen on the following picture(14) t u u u u u ( u + u + u + u ) u ( u + u ) WWWWWWWWWWiiiiiiiiii t u u t u u u ( u + u ) uuuu 66666666 t u u hhhhhhhhhhhhh MMMMMM t = 1. Each F T is a sum ofoperators F σ (sum over all σ such that the decreasing tree of σ − has shape T ). Thiscan be used as in [16] to derive the hook-length formula for binary trees. Example 3.5.
The moulds associated to planar binary trees are related to the solu-tion of the quadratic differential equation(15) dxdt = b ( x, x ) , x (0) = 1where the bilinear map b ( x, y ) is assumed to have an integral representation of thetype(16) b ( x, y ) = Z x u b u y u t u − dµ ( u ) = ( x ∗ b ∗ y )( t )the convolution ∗ being defined by(17) ( x ∗ y )( t ) = Z x u y u t u − dµ ( u ) F. CHAPOTON, F. HIVERT, J.-C. NOVELLI, AND J.-Y. THIBON and b ( t ) = b (1 , t ). This can be recast in the form(18) x = 1 + B ( x, x )where(19) B ( x, y ) = Z t b ( x, y )( τ ) dτ , so that(20) x = 1 + B (1 ,
1) + B ( B (1 , ,
1) + B (1 , B (1 , · · · = X T ∈ CBT B T (1)where CBT is the set of (complete) binary trees, and for a tree T , B T ( a ) is the resultof evaluating the expression formed by labeling by a the leaves of T and by B itsinternal nodes. Then, the term B T (1) in the binary tree solution is F T [ b ]. Example 3.6.
The mould [6, (103)](21) y p,q ( u , . . . , u n ) = u p u · · · u n ( u + · · · + u n )(sum over all binary trees of type ( p, q )) corresponds to the operator(22) Y p,q [ h (1) , . . . , h ( n ) ] = Z t H (1) ( τ ) · · · H ( p − ( τ ) h ( p ) ( τ ) H ( p +1) ( τ ) · · · H ( n ) ( τ ) dτ . Example 3.7.
The mould
T Y defined by [6, (104)](23)
T Y n = n X i =1 α i − y i,n − i corresponds to the operator(24) F [ h ] = Z t (1 − αH ( τ )) − h ( τ )(1 − H ( τ )) − dτ . When h is scalar (the h u commute), this reduces to(25) F [ h ] = Z t (1 − αH ( τ )) − (1 − H ( τ )) − dH ( τ ) = 11 − α log (cid:18) − αH ( t )1 − H ( t ) (cid:19) . Example 3.8.
The mould [6, (106)](26) n X i =1 iy i,n − i corresponds to the operator(27) F [ h ] = Z t (1 − H ( τ )) − h ( τ )(1 − H ( τ )) − dτ . When h is scalar, this reduces to(28) F [ h ] = Z t (1 − H ( τ )) − (1 − H ( τ )) − dH ( τ ) = H ( t )(2 − H ( t ))2(1 − H ( t )) . N OPERATIONAL CALCULUS FOR THE
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Example 3.9.
The following modified mould(29) n X i =1 [ i ] q y i,n − i , where [ i ] q is the quantum number 1 + q + · · · + q i − , corresponds to the operator(30) F [ h ] = Z t (1 − qH ( τ )) − (1 − H ( τ )) − h ( τ )(1 − H ( τ )) − dτ . When h is scalar, this reduces to(31) F [ h ] = Z t (1 − qH ( τ )) − (1 − H ( τ )) − dH ( τ ) = 11 − q (cid:18) H ( t )1 − H ( t ) − q − q log (cid:18) − H ( t )1 − qH ( t ) (cid:19)(cid:19) . Example 3.10.
The Connes-Moscovici series ([6, (109)]) is given by the mould(32) 1 n ! n X k =1 ( − n − k (cid:18) nk (cid:19) ky k,n − k and the corresponding operator is(33) F [ h ] = Z t e H ( τ ) h ( τ ) e − H ( τ ) dτ , which reduces to H ( t ) in the scalar case. Example 3.11.
From the Solomon Lie idempotent, one can define the followingmould(34) 1 n X σ ∈ S n ( − d ( σ ) (cid:18) n − d ( σ ) (cid:19) − f σ . Its output is the logarithm of U ( t ) as defined by (7). It is also called the first Eulerianidempotent.The q -deformation obtained in [18] yields a one-parameter family of moulds(35) 1 n X σ ∈ S n ( − d ( σ ) (cid:20) n − d ( σ ) (cid:21) − q maj ( σ ) − ( d ( σ )+12 ) f σ , where (cid:20) n − d ( σ ) (cid:21) is a quantum binomial coefficient. Example 3.12.
The mould(36) D n = n − X i =0 ( − i u u u · · · u ..i ) u ..n ( u i +1 ..n · · · u n − n u n )(sum over trees of the form (left comb) ∧ (right comb)) corresponds to Dynkin’s idem-potent, more precisely(37) D n [ h ] = Z ∆ n [ ... [ h ( t ) , h ( t )] , h ( t )] ..., h ( t n )] dµ ( t ) · · · dµ ( t n ) . F. CHAPOTON, F. HIVERT, J.-C. NOVELLI, AND J.-Y. THIBON
Example 3.13.
The mould
P O , defined in [6, (113)] by(38)
P O n = 1 u n Y i =2 u + · · · + u i − + qu i u i ( u + · · · + u i )can be decomposed on the permutations f σ as(39) P O n = X σ ∈ S n q s ( σ − ) − f σ , where s ( σ ) is the number of saillances of σ , i.e. , the number of i such that σ i > σ j for all j < i . This statistics has the same distribution as the number of cycles. Example 3.14.
A mould is alternal if and only if it satisfies(40) F [ h + h ] = F [ h ] + F [ h ]whenever h and h commute. Typically, H is a Lie algebra and F takes its valuesin a completion of U ( H ). Then, F is alternal if and only if it preserves primitiveelements. For example, (34), (35) and (36) are alternal. Similarly, F is symmetral ifit maps primitive elements to group-like elements. Otherwise said,(41) F [ h + h ] = F [ h ] · F [ h ]as soon as h and h commute. Example 3.15.
The dendriform products ≺ and ≻ are given by(42)( F ≻ G )[ h ] = Z t F [ h ]( τ ) · ddτ G [ h ]( τ ) dτ , ( F ≺ G )[ h ] = Z t ddτ F [ h ]( τ ) · G [ h ]( τ ) dτ . On permutational moulds f σ , these coincide with the half-shifted shuffles, e.g. , 312 ≺
12 = (31 45) · Example 3.16.
The preLie product F x G = F ≻ G − G ≺ F is given by(43) F x G [ h ] = Z t [ F [ h ] , G ′ [ h ]]( τ ) dτ , where G ′ [ h ] denotes the derivative with respect to τ . On this expression it is clearthat if h is primitive, so is F x G [ h ] if F and G are alternal.4. Operadic operations on operators
The i th operadic composition of two homogeneous moulds f m and g n , as definedin[6], corresponds to the operator whose polarization is(44) F m ◦ i G n [ h (1) , . . . , h ( i − ; h ( i ) , . . . , h ( i + n − ; h ( i + n ) , . . . , h ( m + n − ]= F m [ h (1) , . . . , h ( i − ; ddt G n [ h ( i ) , . . . , h ( i + n − ]; h ( i + n ) , . . . , h ( m + n − ] . It follows from this description that the linear span of the F σ is stable under theseoperations, hence form a suboperad. N OPERATIONAL CALCULUS FOR THE
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Example 4.1.
According to the definition of [6],(45) f ◦ f = 1 u ( u + u ) ( u + u + u + u ) u = f + f + f . and(46) F ◦ F [ h (1) , h (2) , h (3) , h (4) ] = F [ h (1) , ddt F [ h (2) , h (3) ] , h (4) ] , where(47) F [ h (2) , h (3) ] = Z t dt Z t dt h (2) ( t ) h (3) ( t )has as derivative, evaluated at t (48) F [ h (2) , h (3) ] ′ ( t ) = Z t dt h (2) ( t ) h (3) ( t ) . When plugged into F , with the shifts 312 → Z ∆ ( t,t ) h (1) ( t ) h (2) ( t ) h (3) ( t ) h (4) ( t ) dt dt dt dt where the integration domain decomposes as(50)∆ ( t, t ) := { < t < t < t < t ; 0 < t < t } = ∆ ( t ) ∪ ∆ ( t ) ∪ ∆ ( t ) , as expected.To give the general rule, it is sufficient to compute(51) F Id m ◦ i F Id n = X F σ where the sum is over permutations σ in the shuffle(52) ((1 , . . . , i −
1) ( i, . . . , i + n − · ( i + n − , . . . , m + n − . Example 4.2. F ◦ F = F (53) F ◦ F = F + F + F (54) F ◦ F = F + F + F + F + F + F (55)This operad is anticyclic. It is in fact isomorphic to Zinbiel (cf. [21]), up to mirrorimage of permutations. The action of the ( n + 1)-cycle γ on a homogenous mould f ( u , . . . , u n ) of degree n is defined by(56) γf ( u , . . . , u n ) = f ( u , u , . . . , u n , − u − u − · · · − u n ) . The subspace spanned by permutational moulds f σ is stable under the action of γ .Explicitly,(57) γf σ = ( − | v | X τ ∈ u v f τ F. CHAPOTON, F. HIVERT, J.-C. NOVELLI, AND J.-Y. THIBON where the words u and v are defined as follows. Let σ ′ ( i ) = σ ( i ) + 1 mod n , write σ ′ = u w and v = 1 ¯ w (where ¯ w is the mirror image of w ).This follows easily from the product formula for permutational moulds. For exam-ple, γf = − f − f − f − f = − f f (58) γf = f + f + f + f + f + f = f f . (59) Example 4.3.
The operadic preLie product F ◦ G is(60) ( F ◦ G )[ h ] = m X i =1 ( F ◦ i G )[ h ] = DF [ h ]( G ′ [ h ]) , that is, the differential DF [ h ] of F at the point h , evaluated on the vector G ′ [ h ],where, as above, G ′ [ h ] denotes the t -derivative. On this description, it is clear that ◦ preserves alternality. Example 4.4.
The derivation ∂ of [6, (85)] is(61) ( ∂F )[ h ] = DF [ h ](1) := lim ε → F [ h + ε ] − F [ h ] ε , the derivative of F at h in the direction on the constant function 1. On this de-scription, it is easy to check that ∂ is a derivation for the various products. Forexample,(62) ∂ ( F ≻ G )[ h ] = Z t { DF [ h ](1) G ′ [ h ] + F [ h ] DG ′ [ h ](1) } dτ = ( ∂F ≻ G + F ≻ ∂G )[ h ] . For those moulds such that F [ h ] reduces to an analytic function F ( H ) of H in thescalar case, ∂F [ h ] reduces to the derivative of F ( H ) with respect to H . Example 4.5.
The over and under operations are given by(63) (
F/G )[ h (1) , . . . , h ( m + n ) ] = G [ F [ h (1) , . . . , h ( n ) ] h ( n +1) , h ( n +2) , . . . , h ( m + n ) ]and(64) ( F \ G )[ h (1) , . . . , h ( m + n ) ] = F [ h (1) , . . . , h ( n − , h ( n ) G [ h ( n +1) , . . . , h ( m + n ) ]] . Example 4.6.
The ARIT map is(65) ARIT(
F, G )[ h ] = DF [ h ]( G [ h ] h − hG [ h ]) . The ARI map is(66)ARI(
F, G )[ h ] = DF [ h ]( G [ h ] h − hG [ h ]) − DG [ h ]( F [ h ] h − hF [ h ])+ F [ h ] G [ h ] − G [ h ] F [ h ] . Non-crossing trees and non-interleaving forests
In [6], the first author has constructed an operad on the set of non-crossing trees,and formulated a conjecture about the inverse image of non-crossing trees in thedendriform operad. In this section, we prove this conjecture by means of a newpresentation of this operad. The reader is referred to [6] for the background onnon-crossing trees.
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123 45 6 7 89 1011121314
Figure 1.
A non-crossing tree, and the corresponding labeling5.1.
A bijection.Definition 5.1.
A non-interleaving forest is a labeled rooted forest such that the setof labels of any subtree is an interval.
In particular, the labels of each connected component is an interval. A non-interleaving tree is a non-interleaving forest with a single component. Our newpresentation of NCT will be based on non-interleaving trees.Let T be a non-crossing tree. We define a poset P from T as follows. Fist, labeleach diagonal edge of T by the number of the unique open side which it separatesfrom the base, and each side edge by its own number. Then, set i < P j iff the edge i is separated from the base by the edge j . Lemma 5.2. If P is constructed from a non-crossing tree T by the above process, itsHasse diagram F is a non-interleaving forest. Moreover, the correspondence T F is a bijection between non-crossing trees and non-interleaving forests.Proof. The roots of the trees are the labels of the edges having the base on theirexternal sides. The edges α which are on the other sides of the root edges are labeledby disjoint intervals of [1 , n ], and these intervals are the labels of the edges whichare separated from the bases by those α . Conversely, to each vertex v of a non-interleaving forest, on can associate an edge from the left side of min { k | k < P v } tothe right side of max { k | k < P v } . This yields a non-crossing tree mapped to P by theprevious algorithm. Hence, the correspondence is onto. Finally, non-crossing treesand non-interleaving forests have the same grammar, hence in particular the samegenerating series. (cid:3) For example, the non-interleaving forest associated to the non-crossing tree onFigure 1 is (67) 4
MMMMMqqqqq
11 14 yy EE (cid:2)(cid:2) << (cid:2)(cid:2) AAA
12 131 3 5 7 1085.2.
Associated rational functions.
In [6], one associates a rational function f T to a non-crossing tree by the following rule:(68) f T = Y e ∈ E ( T ) e )where E ( T ) is the set of edges of T , and the evaluation of an edge is given by(69) ev( e ) = X i u i , where i runs over the labels of the edges separated from 0 by e .It follows from the above arguments that(70) f T = n Y i =1 P j ≤ P i u j . Lemma 5.3.
The fraction associated to T is the sum of the linear extensions of P : (71) f T = X σ ∈ L ( P ) f σ . Proof. f T is in FQSym , since NCT is a suboperad of Dend,(72) f T = X c σ ( T ) f σ , and c σ ( T ) is the iterated residue of f T T at x σ = 0, x σ = 0 , . . . so that c σ ( T ) = 0 if σ L ( P ), and c σ ( T ) = 1 otherwise. (cid:3) Proof of the conjecture.
Hence, the morphism form the free NCT-algebra onone generator to the free dendriform algebra on one generator
PBT , regarded as asubalgebra of
FQSym , consists in mapping a non-interleaving forest on the sum ofits linear extensions:(73) T P F X σ ∈ L ( P ) F σ = X t ∈ I P t , where P t is the natural basis of PBT , and I a set of binary trees. Conjecture 6.5 of[6] is the following: Theorem 5.4. I is an interval of the Tamari order.Proof. Under this morphism, a forest becomes the product of its connected compo-nents. It is therefore sufficient to prove that the linear extensions of a non-interleavingtree have the required properties. The linear extensions of a tree are computed recur-sively by shuffling the linear extensions of the subtrees of the root and concatenating
N OPERATIONAL CALCULUS FOR THE
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OPERAD 11 the root at the end. By definition of a non-interleaving tree, these form an intervalof the permutohedron, whose minimum avoids the pattern 312 and maximum avoids132. This is a known characterization of Tamari intervals. (cid:3)
Another approach.
One can also start with an operad NIT defined directlyon non-interleaving trees.There are two natural binary operations on non-interleaving trees. Let T and T be two such trees. Let k be the number of vertices of T , and denote by T ′ the resultof shifting the labels of T by k . Define • T ≺ T = grafting of the root of T ′ on the root of T • T ≻ T = grafting of the root of T on the root of T ′ These are two magmatic operations, satisfying the single relation(74) ( x ≻ y ) ≺ z = x ≻ ( y ≺ z ) . The operad defined by this relation has been first considered in [19] under the nameof L-algebra.Consider now the free algebra on one generator. Its monomials can be representedby bicolored complete binary trees, whose internal vertices are colored by ≺ or ≻ .Relation (74) implies that a basis is formed by the trees having no right edge from avertex ≻ to a vertex ≺ . Loday [22] has presented a general method (relying on Koszulduality for quadratic operads) for counting such k colored binary trees avoiding aset Y of edges. Their generating series (with alternating signs) g ( t ) is obtained byinverting (for the composition of power series) the series f ( t ) = − t + kt − | X | t + | X | t − · · · , where X n is the set of trees with n internal nodes whose all edges arein Y . Here k = 2, and there is only one tree, with two internal vertices, having alledges in Y . Hence, f ( t ) = − t + 2 t − t , and we get the sequence A006013 of [26] g ( t ) = − t + 2 t − t + 30 t − t + ... (based non-crossing trees).6. Appendix: Moulds over the positive integers
When the variables u k take only positive integer values, we denote them by i k andwrite f I = f i ,...,i r instead of f ( i , . . . , i r ). This corresponds to the choice(75) dµ ( t ) = X n ≥ δ ( t − n ) . In this case, there is a close connection with the formalism of noncommutative sym-metric functions, which can also represent nonlinear operators on powers series withnoncommuting coefficients.In this appendix, we will give the interpretation of some of the previous examplesin this context, as well as of some new ones. We assume here that the reader isfamiliar with the notation of [18].6.1.
Generating sequences of noncommutative symmetric functions.
Bydefinition,
Sym is a graded free associative algebra, with exactly one generator foreach degree. Several sequences of generators are of common use, some of which beingcomposed of primitive elements, whilst other are sequences of divided powers, so that their generating series is group-like. Each pair of such sequences ( U n ), ( V n ) definestwo moulds, whose coefficients express the expansions of the V n on the U I , and vice-versa. Ecalle’s four fundamental symmetries reflect the four possible combinations ofthe primitive or group-like characteristics.If we denote by L the (completed) primitive Lie algebra of Sym and by G = exp L the associated multiplicative group, we have the following table L → L
Alternal
L → G
Symmetral
G → L
Alternel
G → G
SymmetrelThe characterization of alternal moulds in terms of shuffles is equivalent to Ree’stheorem (cf. [25]): the orthogonal of the free Lie algebra in the dual of the freeassociative algebra is spanned by proper shuffles.
The composition of moulds is the usual composition of the corresponding operators.Since the relationship between two sequences of generators of the same type (dividedpowers or grouplike) can always be written in the form(76) V n ( A ) = U n ( XA ) (or V ( t ) = U ( t ) ∗ σ ( XA )) , where X is a virtual alphabet (commutative and ordered, i.e. , a specialization of QSym ), the composition of alternal or symmetrel moulds can also be expressed bymeans of the internal product.6.2. S n and Λ n : symmetrel. The simplest example just gives the coefficients ofthe inverse of a generic series regarded as λ − t ( A ). It is a symmetrel mould:(77) S n = X I (cid:15) n f I Λ I , f I = ( − n − l ( I ) . S and Ψ : symmetral/alternel. The mould(78) f I = 1 i ( i + i ) . . . ( i + . . . i r )gives the expression of S n over Ψ I :(79) S n = X I (cid:15) n f I Ψ I . Since σ ′ ( t ) = σ ( t ) ψ ( t ), this expresses the solution of the differential equation in termsof iterated integrals(80) σ ( t ) = 1 + Z t dt ψ ( t ) + Z t dt Z t dt ψ ( t ) ψ ( t )+ Z t dt Z t dt Z t dt ψ ( t ) ψ ( t ) ψ ( t ) + · · · = T exp (cid:26)Z t ψ ( s ) ds (cid:27) , N OPERATIONAL CALCULUS FOR THE
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OPERAD 13 or as Dyson’s T -exponential.6.4. An alternal mould: the Magnus expansion.
The expansion of Ψ n in thebasis (Φ K ) is given by(81) Ψ n = X | K | = n ℓ ( K ) X i =1 ( − i − (cid:18) ℓ ( K ) − i − (cid:19) k i Φ K ℓ ( K )! π ( K ) , where π ( K ) = k · · · k r . Using the symbolic notation(82) { Φ i · · · Φ i r , F } = ad Φ i ad Φ i · · · ad Φ i r ( F ) = [Φ i , [Φ i , [ . . . [Φ i r , F ] . . . ]]]and the classical identity(83) e a be − a = X n ≥ (ad a ) n n ! b = { e a , b } , we obtain(84) ψ ( t ) = X n ≥ ( − n ( n + 1)! { Φ( t ) n , Φ ′ ( t ) } = (cid:26) − e − Φ( t ) Φ( t ) , Φ ′ ( t ) (cid:27) which by inversion gives the Magnus formula:(85) Φ ′ ( t ) = (cid:26) Φ( t )1 − e − Φ( t ) , ψ ( t ) (cid:27) = X n ≥ B n n ! (ad Φ( t )) n ψ ( t )the B n being the Bernoulli numbers.6.5. Another alternal mould: the continuous BCH expansion.
The expan-sion of Φ( t ) in the basis (Ψ I ) is given by the series(86) Φ( t ) = X r ≥ Z t dt · · · Z t r − dt r X σ ∈ S r ( − d ( σ ) r (cid:18) r − d ( σ ) (cid:19) − ψ ( t σ ( r ) ) · · · ψ ( t σ (1) ) . Thus, the coefficient of Ψ I = Ψ i · · · Ψ i r in the expansion of Φ n is equal to(87) n Z dt · · · Z t r − dt r X σ ∈ S r ( − d ( σ ) r (cid:18) r − d ( σ ) (cid:19) − t i − σ ( r ) · · · t i r − σ (1) . It is worth observing that this expansion, together with a simple expression ofΨ n in terms of the dendriform operations of FQSym , recently led Ebrahimi-Fard,Manchon, and Patras [9], to an explicit solution of the Bogoliubov recursion forrenormalization in Quantum Field Theory.
Moulds related to the Fer-Zassenhaus series.
The noncommutative powersums of the third kind Z n are defined by(88) σ ( A ; t ) = exp( Z t ) exp( Z t ) . . . exp( Z n n t n ) . . . The first values of Z n are Z = Ψ , Z = Ψ , Z = Ψ + 12 [Ψ , Ψ ] ,Z = Ψ + 13 [Ψ , Ψ ] + 16 [[Ψ , Ψ ] , Ψ ] ,Z = Ψ + 14 [Ψ , Ψ ] + 13 [Ψ , Ψ ] + 112 [[Ψ , Ψ ] , Ψ ] −
724 [Ψ , [Ψ , Ψ ]] + 124 [[[Ψ , Ψ ] , Ψ ] , Ψ ] . This defines interesting alternal moulds. There is no known expression for Z n onthe Ψ I , but Goldberg’s explicit formula (see [25]) for the Hausdorff series gives thedecomposition of Φ n on the basis Z I .The fact that Z n is a Lie series is known as the Fer-Zassenhaus “formula”.6.7. A one-parameter family.
It follows from the characterization of Lie idempo-tents in the descent algebra that P n ( A ; q ) = (1 − q n )Ψ n (cid:16) A − q (cid:17) is a noncommutativepower sum. The corresponding Lie idempotent is(89) ϕ n ( q ) = 1 n X | I | = n ( − d ( σ ) (cid:20) n − d ( σ ) (cid:21) q maj ( σ ) − ( d ( σ )+12 ) σ It specializes to(90) ϕ n (0) = θ n , ϕ n (1) = φ n ϕ n ( ω ) = κ n where ω is a primitive n th root of unity (and ϕ n ( ∞ ) = θ ∗ n ).The nonlinear operator E q [ h ( t )], where h ( t ) = P n ≥ H n t n − , is(91) E q [ h ( t )] = X I c I ( q ) H I t | I | Then,(92) E [ h ( t )] = exp Z t h ( s ) ds = exp H ( t )while E is Dyson’s chronological exponential(93) E [ h ( t )] = T exp Z t h ( s ) ds = 1 + Z t dt h ( t ) + Z t dt Z t dt h ( t ) h ( t ) + · · · N OPERATIONAL CALCULUS FOR THE
MOULD
OPERAD 15
Another one-parameter family.
In [18], it is proved that there exists a uniquesequence π n ( q ) of Lie idempotents which are left and right eigenvectors of σ ((1 − q ) A )for the internal product:(94) σ ((1 − q ) A ) ∗ π n ( q ) = π n ( q ) ∗ σ ((1 − q ) A ) = (1 − q n ) π n ( q )These elements have the following specializations:(95) π n (1) = Ψ n n , n K n ( ζ ) , π n (0) = 1 n Z n . In particular, the associated alternal moulds provide an interpolation between the T -exponential and the Fer-Zassenhaus expansion. Acknowledgements
This work has been partially supported by Agence Nationale de la Recherche, grantANR-06-BLAN-0380
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The On-Line Encyclopedia of Integer Sequences ∼ njas/sequences/(F. Chapoton) Institut Camille Jordan, Universit´e Claude Bernard Lyon 1, F-69622Villeurbanne Cedex, FRANCE (F. Hivert)
LIFAR, Universit´e de Rouen, 76801 Saint-Etienne-du-Rouvray Cedex,FRANCE (J.-C. Novelli, J.-Y. Thibon)
Institut Gaspard Monge, Universit´e Paris-Est, 77454Marne-la-Vall´ee Cedex 2, FRANCE
E-mail address , Fr´ed´eric Chapoton: [email protected]
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E-mail address , Jean-Christophe Novelli: [email protected]
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