aa r X i v : . [ m a t h . N T ] J un MOTIVIC MULTIPLE ZETA VALUES AND THE BLOCKFILTRATION
ADAM KEILTHY
Abstract.
We extend the block filtration, defined by Brown based on the work ofCharlton, to all motivic multiple zeta values, and study relations compatible with thisfiltration. We construct a Lie algebra describing relations among motivic multiplezeta values modulo terms of lower block degree, proving Charlton’s cyclic insertionconjecture in this structure, and showing the existence of a ‘block shuffle’ relation, adihedral symmetry, and differential relation. Introduction
Consider the Tannakian category MT ( Z ) of mixed Tate motives over Spec( Z ), withGalois group G MT ( Z ) . It is well known [8] that this group decomposes as a semidirectproduct of the multiplicative group and a pro-unipotent group, whose graded Lie algebrais non-canonically isomorphic to a free Lie algebra with generators in odd weight g m ∼ = Lie[ σ , σ , . . . ]called the motivic Lie algebra. There exists a non-canonical injection i : { σ k +1 } k ≥ → Q h e , e i , allowing us to consider elements of g m as noncommutative polynomials. ThisLie algebra has ties to the Grothendieck-Teichmuller group [14], associators [9], andmultiple zeta values [3], and injects into Racinet’s double shuffle Lie algebra dmr [14].This Lie algebra is known to be dual to (motivic) multiple zeta values, in the followingsense: in [11], Goncharov defines a bialgebra A of unipotent de Rham multiple zetavalues, arising from the fundamental groupoid of P \ { , , ∞} , which act as formalanalogues to multiple zeta values (modulo ζ (2)), endowed with a coproduct. Brown [3]extends this definition to a comodule H over the graded ring of affine functions overG MT ( Z ) , called motivic multiple zeta values. This algebra, along with its coaction∆ : H → A ⊗ H encodes all motivic relations among multiple zeta values. We have that A = H / ( ζ m (2)),where ζ m (2) is the motivic analogue of ζ (2). Taking the Lie coalgebra of indecomposablesof A , we obtain the dual of g m . Remark 1.1.
We take our convention as identifying e i ↔ dzz − i . As such, elements of g m describe relations among iterated integrals, rather than multiple zeta values. As such,our results are ‘depth-signed’. The Mathematics Institute, University of Oxford, Oxford, United Kingdom
E-mail address : [email protected] . rom this, we can see that g m inherits two filtrations arising from filtrations on motivicmultiple zeta values. The multiple zeta value (MZV) ζ ( n , . . . , n r ) := X ≤ k In addition to the weight and depth filtrations, we will define a ‘block filtration’ on(motivic) multiple zeta values, arising from the work of Charlton [7]. In his thesis,Charlton defines the block decomposition of a word in two letters { x, y } as follows.Begin by defining a word in { x, y } to be alternating if it is non empty and has nosubsequences of the form xx or yy . There are exactly two alternating words of any givenlength: one beginning with x and one beginning with y . Charlton shows that everynon-empty word w ∈ { x, y } × can be written uniquely as a minimal concatenation ofalternating words. In particular, he defines the block decomposition w = w w . . . w k asthe unique factorisation into alternating words such that the last letter of w i equals thefirst letter of w i +1 .We can use this to define a degree function on words in two letters. efinition 2.1. Let w ∈ { x, y } × be a word of length n , given by w = a . . . a n . Defineits block degree deg B ( w ) to be one less than the number of alternating words in its blockdecomposition. Equivalently, definedeg B ( w ) := { i : 1 ≤ i < n such that a i = a i +1 } Remark 2.2. Note that, unlike depth, the block degree of a word is preserved by theduality anti-homomorphism mapping e i e − i , induced by the automorphism z − z of P \ { , , ∞} .We can then define an increasing filtration on Q h e , e i by B n Q h e , e i := h w : deg B ( w ) ≤ n i Q which, following the suggestion of Brown [2], when restricted to a filtration on e Q h e , e i e induces a filtration on motivic multiple zeta values B n H := h ζ m ( w ) : w = e ue , deg B ( w ) ≤ n i Q . Brown goes on to show the following. Proposition 2.3 (Brown) . Let G dR MT ( Z ) denote the de Rham motivic Galois group ofthe category MT ( Z ) , and let U dR MT ( Z ) denote its unipotent radical. Then B n is stableunder the action of G dR MT ( Z ) , and U dR MT ( Z ) acts trivially on gr B H . Equivalently ∆ r ( B n H ) ⊂ O ( U dR MT ( Z ) ) ⊗ B n − H where ∆ r ( x ) := ∆( x ) − x ⊗ − ⊗ x is the reduced coproduct. Corollary 2.4 (Brown) . The block filtration induces the level filtration on the subspacespanned by the Hoffman motivic multiple zeta values ζ m ( n , . . . , n r ) , with n i ∈ { , } ,where the level is the number of indices equal to .Proof. The word corresponding to ( n , . . . , n r ), with n i ∈ { , } of level m has exactly m occurrences of the subsequence e e and none of e e . Therefore, its block degree isexactly m . (cid:3) As a corollary to both this and Brown’s proof that the Hoffman motivic multiple zetavalues form a basis of H , we obtain the following. Corollary 2.5 (Brown) . Every element in B n H of weight N can be written uniquely asa Q -linear combination of motivic Hoffman elements of weight N and level at most n .Additionally X m, n ≥ dim gr B m H n s m t n = 11 − t − st where H n denotes the weight n piece of H . However, trying to naively extend this filtration by B n H = h ζ m ( w ) : deg B ( w ) ≤ n i Q we find that the associated graded gr B H becomes nearly trivial. If we instead extendthe filtration as follows, we obtain a much more interesting structure. efinition 2.6. We define the block filtration of Q h e , e i by B n Q h e , e i := h w : deg B ( e we ) ≤ n i Q . This induces the block filtration of motivic multiple zeta values B n H := h ζ m ( w ) : deg B (0 w ≤ n i Q . This filtration agrees with our earlier definition if we restrict to w ∈ e Q h e , e i e , butthe associated graded remains interesting. Proposition 2.7. ∆ r B n H ⊂ n − X k =1 B k H ⊗ B n − k H Proof. We will in fact show a stronger statement, that ∆ is graded for block degree atthe level of words. Let I := h I f (0; w ; 1) : w ∈ { , } × i Q be the vector space spanned byformal symbols, with natural projection I → H , I f (0; w ; 1) I m (0; w ; 1)and, similarly, a natural projection I → A .Recall that the motivic coaction is given by the formula∆ I m ( a ; a , . . . , a n ; a n +1 ) : = X i / A > A > . Note that these lift tocoactions I → I ⊗ I , by calculating these purely symbolically.Define I n := h I f (0; w ; 1) : deg B (0 w 1) = n i Q . It is sufficient to show that ∆ I n ⊂ P ni =0 I i ⊗ I n − i , as the result follows upon composition with the necessary projections.In fact, it suffices to show that D r +1 I n ⊂ n X i =0 I i ⊗ I n − i . Now, consider I f (0; w ; 1), w a word in { , } such that deg B (0 w 1) = n . Then we candecompose 0 w b b . . . b n +1 into alternating blocks, and consider the action of D n +1 on I f ( b . . . b n +1 ). All terms in D n +1 I f ( b . . . b n +1 ) will be of the formI f ( x ; b ′′ i b i +1 . . . b ′ i + j ; y ) ⊗ I f ( b . . . b i − b ′ i xyb ′′ i + j b i + j +1 . . . b n +1 ) or some 1 ≤ i ≤ n + 1, where b i = b ′ i xb ′′ i , b i + j = b ′ i + j yb ′′ i + j . For the left hand term to benon-zero, we must have x = y , and so we seedeg B ( xb ′′ i b i +1 . . . b ′ i + j y ) = j, deg B ( b . . . b ′ i xyb ′′ i + j . . . b n +1 ) = n − j, by counting the blocks. Thus, we get that the total block degree of any term in thecoproduct is n , and the result follows. (cid:3) Corollary 2.8. The block filtration on Q h e , e i induces the coradical filtration on H . Corollary 2.9. The (linearised) Ihara action ◦ : Q h e , e i ⊗ Q h e , e i → Q h e , e i isgraded for block degree.Proof. The Ihara action is dual to the motivic coaction. As this proof shows the coactionto be, at the level of words, graded for block degree, the claim follows immediately. Onecan also show this directly via the recursive formula [5] for the linearised Ihara action. (cid:3) We also recall a short observation due to Charlton [7]. Lemma 2.10. Let w = w . . . w n be a word in { , } × of length n , with deg B ( w ) = b .Then I m ( w ) = 0 if b ≡ w + 1 (mod 2). Remark 2.11. This provides a natural analogue of the depth parity theorem [4]. Proposition 2.12. Suppose σ ∈ ls is of weight N and depth d . Then, if N and d areof opposite parity, σ = 0 . That is, there are no non-trivial solutions to the lineariseddouble shuffle equations with weight and depth of opposite parity. With Proposition 3.5, we obtain a similar corollary to the final conclusion of thefollowing corollary. Corollary 2.13. For a solution to the double shuffle equations mod products φ ∈ dmr ,of weight N , the depth d + 1 N (mod 2) components are uniquely determined by thelower depths. In particular, σ n +1 is uniquely determined in depths 1 and 2. Specifically, σ n +1 is uniquely determined in block degree 1 and 2.3. Block-graded multiple zeta values and an encoding of relations As the block filtration is motivic and invariant under the duality arising from thesymmetry z − z of P \ { , , ∞} , we can consider the associated graded algebragr B A := L ∞ n =0 B n A / B n − A . We follow the example of Brown’s depth graded multiplezeta values [4]. Remark 3.1. In the following it is important to keep in mind that we are identifying e i ↔ dzz − i , and, as such, elements of g m describe relations among iterated integrals, ratherthan multiple zeta values. As such, our results are ‘depth signed’ compared to standardnotation. Definition 3.2. Define B n Q h e , e i := h w : deg B ( e we ) ≥ n i Q and definegr B g m := ∞ M n =0 B n g m / B n +1 g m here we identify B n g m / B n +1 g m with its image in B n Q h e , e i / B n +1 Q h e , e i , equippedwith the block graded Ihara bracket. Definition 3.3. If deg B ( e we ) = n , define I b (0; w ; 1) to be the image of I a (0; w ; 1) in B n A / B n − A . Similarly, define I bl (0; w ; 1) to be the image of I l (0; w ; 1) in B n L / B n − L .Define ζ b and ζ bl similarly. Definition 3.4. Fix an embedding of { σ , σ , . . . } ֒ → Q h e , e i . We define the blockgraded generators { p k +1 } k ≥ to be the image of the generators { σ k +1 } k ≥ of g m in B Q h e , e i / B Q h e , e i . We define the bigraded Lie algebra bg to be the Lie algebragenerated by p k +1 and the Ihara bracket.One of the challenges in studying g m is that we have an ambiguity in our representationof the generators: σ k +1 is unique only up to addition of another element of weight 2 k +1.Its depth one part is canonical, so Brown’s depth graded Lie algebra avoids this issue.We find similar success here. Proposition 3.5. The generators p k +1 of bg are canonical, i.e. independent of ourchoice of embedding of generators { σ k +1 } ֒ → Q h e , e i .Proof. Let σ k +1 , σ ′ k +1 ∈ Q h e , e i be two choices of generator for g m in weight 2 k + 1.We must have σ k +1 − σ ′ k +1 ∈ { g m , { g m , g m }} . Corollary 2.9 tells us that the Ihara action is compatible with the block filtration, andso { g m , { g m , g m }} ⊂ B g m and therefore p k +1 − p ′ k +1 = 0 . (cid:3) Note that we can still define a concept of depth on bg as before. We define the depthof a word w to be d ( w ), and induce a decreasing filtration on bg via its embedding bg ֒ → Q h e , e i . It is interesting here that depth grading gives canonical generators indepth 1, while block grading gives p k +1 consisting only of terms of depth k or k + 1. Lemma 3.6. p k +1 contains only depth k and k + 1 terms.Proof. Suppose w is a word of block degree 1 and weight 2 k + 1. Then e we has twoblocks and hence contains exactly one of e or e . In the first case, the number of e must be exactly half 2 k + 1 − 1, i.e. k . In the second case, the number of e mustsimilarly be k and hence the number of e is k + 1. (cid:3) Theorem 3.7. bg is freely generated by { p k +1 } k ≥ as a Lie algebra.Proof. We have a bijection between the generators of g m and of bg , and Corollary 2.9tells us that the Ihara action is graded for block degree. Thus, we can write an element { p k +1 , { . . . , { p k b − +1 , p k b +1 } , . . . }} as the image of { σ k +1 , { . . . , { σ k b − +1 , σ k b +1 } , . . . }} n B b g m / B b +1 g m . Hence, we have a relation in bg if and only if the corresponding sum ofterms is 0 in gr B g m . Indeed, we have an injective Lie algebra homomorphism bg ֒ → gr B g m induced by the bijection { σ k +1 } k ≥ ↔ { p k +1 } k ≥ . Now, as gr B g m is dual to gr B L , theexistence of relations in bg implies the existence of additional relations in gr B L . To beprecise, we must have thatdim gr B n L N < dim h I l ( w ) | deg B ( w ) = n, | w | = N i Q . Then, by the proof of Theorem 7.4 in [3], we know that the right hand side has is spannedby { ζ a ( k , . . . , k r ) } , where k i ∈ { , } , and k i = 3 exactly n times and k + · · · + k r = N .In particular, it has a basis given by ζ a ( k , . . . , k r ) such that ( k , . . . , k r ) is a Lyndonword with respect to the order 3 < 2. This basis, called the Hoffman-Lyndon basis,forms a spanning set for gr B n L N . Thus,dim gr B n L N < dim h I a ( w ) | deg B ( w ) = n, | w | = N i Q which implies that there is a sum of Hoffman-Lyndon elements of weight N with n threes that can be written as a sum of Hoffman-Lyndon elements of weight N withfewer threes. However, the Hoffman-Lyndon elements of weight N form a basis of L N ,and, so, no such relation can exist. Thus, we must have that L ≡ gr B L as they haveequal dimensions, and hence, gr B g m ≡ g m . This implies gr B g m and bg are both freelygenerated and isomorphic. (cid:3) Remark 3.8. While both Brown’s dg and our bg have canonical generators, Theorem3.7 tells us that bg is free, while there exist relations in dg , and hence ‘exceptional’generators are needed, first appearing in depth four. These relations are shown tohave a somewhat mysterious connection to modular forms by Pollack [13], and this hasbeen further explored by Baumard and Schneps [1]. However, it is a computationallychallenging task, and suggests that ‘depth graded’ multiple zeta values may not be themost natural choice of object to study.4. Polynomial representations We now reframe this Lie algebra in terms of commutative polynomials, similarly toBrown [5][6] and ´Ecalle [10], as follows.Recall that Charlton shows that every word w ∈ { e , e } × can be written uniquely asa sequence of alternating blocks [7]. In doing so, he establishes a bijectionbl : { e , e } × \ {∅} → ∪ ∞ n =1 { , } × N n w ( ǫ ; l , l . . . , l n )where ǫ defines the first letter of w , and l , . . . , l n describe the length of the alternatingblocks. Example 4.1. e e e e e e e e (0; 3 , , ,e e e e e e e e e (1; 1 , , , e can use this to define a vector space isomorphism by π bl : Q h e , e i \ { Q · } → ∞ M n =1 x Q [ x , . . . , x n ] x n w x l . . . x l n n (4.1)where bl( e we ) = (0; l , . . . , l n ).In this formulation, a word of block degree n and weight N ≥ n + 1 variables of degree N + 2. From this point on, we shall freely identifyelements of bg with their images under this isomorphism. Proposition 4.2. The projections of the depth-signed σ k +1 ∈ g m onto their block degreeone part are given by p k +1 ( x , x ) = q k +1 ( x , x ) − q k +1 ( x , x ) where q k +1 ( x , x ) = k X i =1 (cid:20)(cid:18) k i (cid:19) − (cid:18) − k (cid:19) (cid:18) k k + 1 − i (cid:19)(cid:21) x i +11 x k +2 − i − x x k +22 and σ k +1 have been normalised to correspond to ( − k ζ (2 k + 1) .Proof. We will compute the block degree 1 part of σ k +1 consisting of terms containingan e . This will give q k +1 . That p k +1 ( x , x ) = q k +1 ( x , x ) − q k +1 ( x , x ) followsfrom duality. In terms of e , e , we have q k +1 = k X i =0 c i ( e e ) i e ( e e ) k − i , where ζ m ( { } i − , , { } k − i ) = αc i ζ m (2 k + 1) (mod ζ m (2)), for i > α ∈ Q ,and c is obtained via shuffle regularisation [3].Shuffle regularisation of e e . . . e tells us that c + 2 k X i =1 c i = 0 . Next, from the work of Zagier [16], ζ ( { } a , , { } b ) = 2 a + b +1 X r =1 ( − r (cid:20)(cid:18) r a + 2 (cid:19) − (1 − r ) (cid:18) r b + 1 (cid:19)(cid:21) ζ ( { } a + b − r +1 ) ζ (2 r +1) . Brown then shows in [3] Theorem 4.3 that this lifts to an identity among motivic multiplezeta values. Considered modulo ζ m (2), we find ζ m ( { } i − , , { } k − i ) = 2( − k (cid:20)(cid:18) k i (cid:19) − (cid:18) − k (cid:19) (cid:18) k k + 1 − i (cid:19)(cid:21) ζ m (2 k + 1) , and thus, we can take c i = h(cid:0) k i (cid:1) − (1 − k ) (cid:0) k k +1 − i (cid:1)i for i > 0. The result thenfollows. (cid:3) Computing these sums explicitly, we obtain the following theorem. heorem 4.3. p k +1 ( x , x ) = x x ( x − x ) (cid:18) (1 − k +1 )( x + x ) k − ( x − x ) k k (cid:19) . With this in mind, we can provide a characterisation of these generators in terms ofpolynomial equations. Corollary 4.4. The polynomial p k +1 ( x , x ) is, up to rescaling, the unique homoge-neous polynomial p ( x , x ) of degree k + 3 such that p ( x , 0) = p (0 , x ) = p ( x , x ) + p ( x , x ) = 0 , and, defining r ( x , x ) := p ( x ,x ) x x ( x − x ) , satisfying r (0 , x ) = 2 r ( x, − x ) , and (cid:18) ∂∂x (cid:19) r ( x , x ) = (cid:18) ∂∂x (cid:19) r ( x , x ) . Proof. The condition p ( x , 0) = p (0 , x ) = p ( x , x ) + p ( x , x ) = 0 suggests we canwrite p ( x , x ) = x x ( x − x ) r ( x , x ). Letting u = x + x , and v = x − x , we canrewrite (cid:18) ∂∂x (cid:19) r ( x , x ) = (cid:18) ∂∂x (cid:19) r ( x , x ) ⇔ ∂ r∂u∂v ( u, v ) = 0 , which has polynomial solution, homogeneous of degree (2 k + 3) − kr ( u, v ) = αu k + βv k which is to say r ( x , x ) = α ( x + x ) k + β ( x − x ) k . Finally, the condition r (0 , x ) = 2 r ( x, − x )gives ( α + β ) x k = 2 k +1 βx k , and hence α = − (1 − k +1 ) β, giving the desired result. (cid:3) We can provide an exact polynomial formula for the Ihara action. Recall that we havechosen g m to differ from Brown’s by sending e 7→ − e , and so this is only accurate for‘depth-signed’ elements. We delay the proof of this until later. heorem 4.5. For (depth-signed) elements of the motivic Lie algebra, the Ihara actionis given at the level of block-polynomials by ( f ◦ g )( x , . . . , x m + n − ) = ( − ( m +1)( n +1) n X i =1 f ( x i , x i +1 , . . . , x i + m − ) x i − x i + m − × (cid:18) x i g ( x , . . . , x i − , x i , x i + m , . . . , x m + n − ) − x i + m − g ( x , . . . , x i − , x i + m − , . . . , x m + n − ) (cid:19) . (4.2) 5. Relations arising in the polynomial representation We find several relations arising naturally in the polynomial representation which arepreserved by the Ihara action, and dual to relations in gr B L . We start by once againshowing that duality is indeed preserved by our formula. Proposition 5.1. For all f ( x , . . . , x n ) ∈ bg , f ( x , . . . , x n ) = ( − n +1 f ( x n , . . . , x ) . Proof. It suffices to show that this holds for p k +1 , and that, if this holds for f, g ∈ bg ,then it holds for f ◦ g . The former holds by definition of p k +1 . To see the latter, notethat( f ◦ g )( x m + n − , . . . , x ) = ( − ( m +1)( n +1) n X i =1 f ( x m + n − i , x m + n − i − , . . . , x n +1 − i ) x m + n − i − x n +1 − i × (cid:18) (1 + x n +1 − i x m + n − i ) g ( x m + n − , . . . , x m + n − i +1 , x m + n − i , x n − i , . . . , x ) − (1 + x m + n − i x n +1 − i ) g ( x m + n − , . . . , x m + n − i +1 , x n +1 − i , . . . , x ) (cid:19) = ( − ( m +1)( n +1) n X i =1 ( − m f ( x n +1 − i , x n +2 − , . . . , x m + n − i ) x n +1 − i − x m + n − i × (cid:18) ( − n +1 (1 + x n +1 − i x m + n − i ) g ( x , . . . , x n − i , x m + n − i , x m + n − i , . . . , x m + n − ) − ( − n +1 (1 + x m + n − i x n +1 − i ) g ( x , . . . , x n +1 − i , x m + n − i +1 , . . . , x m + n − ) (cid:19) = ( − m + n ( − ( m +1)( n +1) n X i =1 f ( x n +1 − i , x n +2 − , . . . , x m + n − i ) x n +1 − i − x m + n − i × (cid:18) (1 + x m + n − i x n +1 − i ) g ( x , . . . , x n +1 − i , x m + n − i +1 , . . . , x m + n − ) − (1 + x n +1 − i x m + n − i ) g ( x , . . . , x n − i , x m + n − i , x m + n − i , . . . , x m + n − ) (cid:19) = ( − m + n ( f ◦ g )( x , . . . , x m + n − ) , nd hence, the duality relation is preserved by the Ihara bracket. (cid:3) We can similarly prove Charlton’s cyclic insertion conjecture, up to terms of lowerblock degree. While this has been verified in upcoming work due to Hirose-Sato, in thisformulation, it is merely a consequence of the Ihara action, allowing for a significantlysimpler proof. We will instead show that a more general relation holds, of which cyclicinsertion is a corollary. These are the ‘block shuffle’ relations. Definition 5.2. For any 1 ≤ r ≤ n , define the shuffle setSh n,r = { σ ∈ S n | σ − (1) < . . . < σ − ( r ); σ − ( r + 1) < . . . < σ − ( n ) } . Then, for any f ∈ Q [ x , . . . , x n ], define f ( x . . . x r (cid:1) x r +1 . . . x n ) := X σ ∈ Sh n,r f ( x σ (1) , x σ (2) , . . . , x σ ( n ) ) . Theorem 5.3. For any f ( x , . . . , x n ) ∈ bg , and any ≤ r < n , we have f ( x x . . . x r (cid:1) x r +1 . . . x n ) = 0 . Proof. For p k +1 , this is equivalent to p ( x , x ) + p ( x , x ) = 0, given by Proposition 5.1.Then, as the Ihara action is associative, it in fact suffices to show that( f ◦ g )( x . . . x r (cid:1) x r +1 . . . x n +1 ) = 0for all f = p k +1 ( x , x ), g ( x , . . . , x n ) ∈ bg .We write ( f ◦ g )( x . . . x r (cid:1) x r +1 . . . x n +1 ) as X σ ∈ Sh n +1 ,r n X i =1 f ( x σ ( i ) , x σ ( i +1) ) x σ ( i ) − x σ ( i +1) × (cid:18) g ( x σ (1) , . . . , x σ ( i ) , x σ ( i +2) , . . . , x σ ( n +1) ) x σ ( i ) − g ( x σ (1) , . . . , x σ ( i − , x σ ( i +1) , . . . , x σ ( n +1) ) x σ ( i +1) (cid:19) . his sum splits as follows X σ ∈ Sh n +1 ,r r − X i =1 f ( x σ ( i ) , x σ ( i +1) ) x σ ( i ) − x σ ( i +1) × (cid:18) g ( x σ (1) , . . . , x σ ( i ) , x σ ( i +2) , . . . , x σ ( n +1) ) x σ ( i ) − g ( x σ (1) , . . . , x σ ( i − , x σ ( i +1) , . . . , x σ ( n +1) ) x σ ( i +1) (cid:19) + X σ ∈ Sh n +1 ,r n X i = r +1 f ( x σ ( i ) , x σ ( i +1) ) x σ ( i ) − x σ ( i +1) × (cid:18) g ( x σ (1) , . . . , x σ ( i ) , x σ ( i +2) , . . . , x σ ( n +1) ) x σ ( i ) − g ( x σ (1) , . . . , x σ ( i − , x σ ( i +1) , . . . , x σ ( n +1) ) x σ ( i +1) (cid:19) + X σ ∈ Sh n +1 ,r such that { σ ( r ) ,σ ( r +1) }6 = { r,r +1 } f ( x σ ( r ) , x σ ( r +1) ) x σ ( r ) − x σ ( r +1) × (cid:18) g ( x σ (1) , . . . , x σ ( r ) , x σ ( r +2) , . . . , x σ ( n +1) ) x σ ( r ) − g ( x σ (1) , . . . , x σ ( r − , x σ ( r +1) , . . . , x σ ( n +1) ) x σ ( r +1) (cid:19) . Denote the first sum by A , the second by B , and the third by C . Now, this sum canbe written uniquely as X ≤ k For any finite sequence of integers l , . . . , l n , and any ≤ r < n , wehave X σ ∈ Sh n,r I bl (( l σ (1) , . . . , l σ ( n ) ) = 0 when considered modulo products.Proof. Using Theorem 3.7, we can consider bg as the dual Lie algebra to the gradedLie coalgebra of indecomposables gr B L , and hence, relations among the coefficients ofelements of bg induce relations among elements of gr B L . Specifically, we define a Q -linearpairing h I bl ( l , . . . , l n ) | x k . . . x k m m i := δ l ,k . . . δ l n ,k n here I bl ( l , . . . , l n ) is the image of I b ( l , . . . , l n ) in gr B L . We have that R is a relationin gr B L if and only if h R | f i = 0 for all f ∈ bg . Hence, as f ( x x . . . x r (cid:1) x r +1 . . . x n ) = 0for all f ∈ bg , we must have that X σ ∈ Sh n,r I bl (( l σ (1) , . . . , l σ ( n ) ) = 0 . (cid:3) Corollary 5.5 (Block graded cyclic insertion) . The cyclic sum X σ ∈ C n I bl ( l σ (1) , l σ (2) , . . . , l σ ( n ) ) = 0 . Proof. It suffices to show that X σ ∈ C n f ( x σ (1) , x σ (2) , . . . , x σ ( n ) ) = 0for all f ∈ bg .Suppose f ∈ bg . Then, Theorem 5.3 implies that the image of f under the followingvector space isomorphism ∞ M n =0 Q [ x , . . . , x n ] ∼ −→ Q h z , z , z , . . . i x i x i . . . x i n n z i z i . . . z i n (5.1)lies in Lie[ z , z , . . . ]. In particular, the image lies in the span of elements of degree atleast 2. Now, we define a linear map C : Q h z , z , . . . i → Q h z , z , . . . i by C ( z i z i . . . z i n ) = X σ ∈ C n z i σ (1) z i σ (2) . . . z i σ ( n ) for a word of length n . Thus, it suffices to show that C ( Z ) = 0 for all Z ∈ Lie[ z , z , . . . ]of degree at least 2.Note, for any monomials X, Y in { z , z , . . . } of degree k, n − k respectively, we have[ X, Y ] = XY − σ ( XY ), for some σ ∈ C n acting by cyclic rotations on words of length n . Thus, C ([ X, Y ]) = C ( XY ) − C ( σ ( XY )) = C ( XY ) − C ( XY ) = 0and so the image of any element of degree at least two in Lie[ z , z , . . . ] is zero, and hence X σ ∈ C n f ( x σ (1) , x σ (2) , . . . , x σ ( n ) ) = 0 (cid:3) Remark 5.6. As in this proof, it can be useful to consider bg as a subspace of the Hopf Q h z , z , z , . . . i , with the standard concatenation product, and a coproduct given by∆ z i = z i ⊗ ⊗ z i . For example, Theorem 5.3 implies elements of bg are primitive forthis coproduct, we immediately obtain Proposition 5.1 as a corollary, by considering theantipode map, i.e. the antihomomorphism z i 7→ − z i . This is an idea explored further inSection ?? . . Shuffle Regularisation The double shuffle relations among iterated integrals are not, in general, compatiblewith the block filtration. However, the regularisation relation obtained by shuffling withan element of weight 1, does respect the block filtration. Theorem 6.1. Let π : Q h e , e i → Q e ⊕ Q e denote the projection map onto weight1 , and let ∆ : Q h e , e i → Q h e , e i ⊗ Q h e , e i be the coproduct defined by ∆( e i ) = e i ⊗ ⊗ e i . The map ∆ := ( π ⊗ id )∆ is compatible with the block filtration: ∆ B n Q h e , e i ⊂ B Q h e , e i ⊗ B n − Q h e , e i . Proof. For w ∈ Q h e , e i every term in ∆ ( w ) is of the form e i ⊗ ¯ w for i ∈ { , } , where¯ w is obtained from w by omitting a letter. The left hand side is of block degree 1. Theright hand side is of higher block degree, if the omitted letter was internal to a block,and of block degree 1 lower than w , if the omitted letter was at the beginning or end ofa block. (cid:3) Thus, we can take the associated graded map of ∆ . Corollary 6.2. gr B (∆ )( bg ) = 0 . Proof. This follows from the work of Brown [3] and Racinet [14], as any element ψ ∈ g m satisfies ∆( ψ ) = 0. (cid:3) In low degree, we can translate this to a statement about elements of bg consideredas polynomials. Example 6.3. For f ( x , x ) ∈ bg and g ( x , x , x ) ∈ bg , we have x ∂f∂x (0 , x ) = f ( x, − x ) ,yz (cid:18) ∂g∂x (0 , y, z ) − ∂g∂x (0 , y, − z ) (cid:19) = y ( g ( y, z, − z ) + g ( − y, z, − z ))+ z ( g ( − y, y, − z ) − g ( − y, y, z )) ,yz (cid:18) ∂g∂x (0 , y, z ) + ∂g∂x (0 , y, − z ) + ∂g∂x ( y, , z ) + ∂g∂x ( y, , − z ) (cid:19) = y ( g ( y, z, − z ) − g ( − y, z, − z )) − z ( g ( − y, y, − z ) + g ( − y, y, z )) . In order to better describe elements of bg , we use the following lemma to transformour polynomial representation. Lemma 6.4. For f ( x , x , . . . , x n ) ∈ bg , we can write f ( x , . . . , x n ) = x . . . x n ( x − x n ) r ( x , . . . , x n ) for some polynomial r ∈ Q [ x , . . . , x n ] . roof. We induct on the number of variables. For n = 2, this follows from Theorem 4.3.Now, suppose this factorisation holds for f ( x , x ) , g ( x , . . . , x n ) ∈ bg . We have { f, g } = n X i =1 f ( x i , x i +1 ) x i − x i +1 × (cid:18) x i g ( x , . . . , x i , x i +2 . . . , x n +1 ) − x i +1 g ( x , . . . , x i − , x i +1 , . . . , x n +1 ) (cid:19) − g ( x , . . . , x n ) (cid:18) x f ( x , x n +1 ) − x n f ( x n , x n +1 ) (cid:19) − g ( x , . . . , x n +1 (cid:18) x f ( x , x ) − x n +1 f ( x , x n +1 ) (cid:19) . Applying our induction hypothesis, we find { f, g } = x . . . x n +1 r f ( x , x ) × (( x − x n +1 ) r g ( x , x , . . . , x n +1 ) − ( x − x n +1 ) r g ( x , . . . , x n +1 ))+ n − X i =2 x . . . x n +1 ( x − x n +1 ) r f ( x i , x i +1 ) × ( r g ( x , . . . , x i , x i +2 . . . , x n +1 ) − r g ( x , . . . , x i − , x i +1 , . . . , x n +1 ))+ x . . . x n +1 r f ( x n , x n +1 ) × (( x − x n ) r g ( x , . . . , x n ) − ( x − x n +1 ) r g ( x , . . . , x n − , x n +1 )) − x . . . x n +1 r g ( x , . . . , x n ) (( x − x n +1 ) r f ( x , x n +1 ) − ( x n − x n +1 ) r f ( x n , x n +1 )) − x . . . x n +1 r g ( x , . . . , x n +1 ) (( x − x ) r f ( x , x ) − ( x − x n +1 ) r f ( x , x n +1 )) . Considering only the terms not immediately divisible by x . . . x n +1 ( x − x n +1 ), wereduce the problem to showing that − x . . . x n +1 ( x − x n +1 ) r f ( x , x ) r g ( x , . . . , x n +1 )+ x . . . x n +1 ( x − x n ) r f ( x n , x n +1 ) r g ( x , . . . , x n )+ x . . . x n +1 ( x n − x n +1 ) r f ( x n , x n +1 ) r g ( x , . . . , x n ) − x . . . x n +1 ( x − x ) r f ( x , x ) r g ( x , . . . , x n +1 )= − x . . . x n +1 ( x − x n +1 ) r f ( x , x ) r g ( x , . . . , x n +1 )+ x . . . x n +1 ( x − x n +1 ) r f ( x n , x n +1 ) r g ( x , . . . , x n )is divisible by x . . . x n +1 ( x − x n +1 ). This is clear, so we are done. (cid:3) Definition 6.5. For f ( x , . . . , x n ) ∈ bg , define the reduced block polynomial to be r ( x , . . . , x n ) := f ( x , . . . , x n ) x . . . x n ( x − x n ) . Define rbg to be the bigraded Q -vector space of reduced block polynomials. Remark 6.6. It may be useful to recall how the various degrees we assign to motiviciterated integrals relate to the reduced block polynomials. A reduced block polynomial ( x , x , . . . , x n ) of degree N corresponds to elements of weight N + n − n − 1. 7. The dihedral action As an immediate corollary to Proposition 5.1 we obtain: Lemma 7.1. For all r ( x , . . . , x n ) ∈ rbg , r ( x n , . . . , x ) = ( − n r ( x , . . . , x n ) . Definition 7.2. We define a Lie algebra structure on rbg via the Lie bracket { r , r } ( x , . . . , x m + n − ) := { f , f } ( x , . . . , x m + n − ) x . . . x m + n − ( x − x m + n − )for r ( x , . . . , x m ) = f ( x ,...,x m ) x ...x m ( x − x m ) , r ( x , . . . , x n ) = f ( x ,...,x n ) x ...x n ( x − x n ) ∈ rbg . We call this thereduced Ihara bracket. It produces a polynomial of degree deg( r ) + deg( r ).We can explicitly compute this, and in the case of r = r ( x , x ), we obtain aparticularly nice formula. Proposition 7.3. For r ( x , x ) , q ( x , . . . , x n − ) ∈ rbg , the reduced Ihara bracket isgiven by { r, q } ( x , . . . , x n ) = n X i =1 r ( x i , x i +1 )( q ( x , . . . , x i , x i +2 , . . . , x n ) − q ( x , . . . , x i − , x i +1 , . . . , x n )) where we consider indices modulo n . Corollary 7.4. r ( x , . . . , x n ) = r ( x , . . . , x n , x ) for r ( x , . . . , x n ) ∈ rbg .Proof. This follows from a simple induction argument, using Lemma 7.1 as our basecase, and the natural cyclic symmetry in Proposition 7.3. (cid:3) Remark 7.5. Corollary 5.5 follows as an immediate corollary to this invariance.With this cyclic invariance, we can write down the general case of the reduced Iharabracket quite succinctly. Corollary 7.6. For r ( x , . . . , x m ) , q ( x , . . . , x n ) ∈ rbg , the reduced Ihara bracket is givenby { r, q } ( x , . . . , x m + n − ) = m + n − X i =1 r ( x i , . . . , x i + m − ) ( q ( x i + m , . . . , x m + n − , x , . . . , x i ) − q ( x i + m − , . . . , x m + n − , x , . . . , x i − )) where the indices are considered modulo m + n − . Thus, we have an action of the dihedral group on rbg , restricting to either the trivialor sign representation on the block graded parts. . A differential relation We additionally obtain a differential relation, generalising the differential relationdefining the generators of bg . Definition 8.1. For n ≥ 2, define the differential operatorD n : Q [ x , . . . , x n ] → Q [ x , . . . , x n ]by D n := Y i ,...,i n − ∈{ , } (cid:18) ∂∂x + ( − i ∂∂x + · · · + ( − i n − ∂∂x n (cid:19) . Theorem 8.2. D n r ( x , . . . , x n ) = 0 for all r ( x , . . . , x n ) ∈ rbg . Proof. We induct on n . For n = 2, this follows from Corollary 4.4. Suppose this holdsfor q ( x , . . . , x n ) ∈ rbg .Next define I n := { M ∈ M n ( µ ) | M i,i = 1 , M i +1 ,j M i,j = M i +1 ,j +1 M i,j +1 } , and L M := n X i =1 M ,i ∂∂x i = ± n X i =1 M j,i ∂∂x i . Note that D n = Q M ∈ I n L M , and thus we have, for r ( x , x ) , q ( x , . . . , x n ) ∈ rbg ,D n +1 { r, q } ( x , . . . , x n +1 ) = n X i =1 D n +1 ( r ( x i , x i +1 ) q ( x i , x i +2 , . . . , x i + n ) − r ( x i , x i +1 ) q ( x i +1 , x i +2 , . . . , x i + n )= n X i =1 X S ⊂ I n +1 ( Y M ∈ S L M ) r ( x i , x i +1 )( Y M ∈ I n +1 \ S L M ) q ( x i , x i +2 , . . . , x i + n ) − n X i =1 X S ⊂ I n +1 ( Y M ∈ S L M ) r ( x i , x i +1 )( Y M ∈ I n +1 \ S L M ) q ( x i +1 , x i +2 , . . . , x i + n )where we have used the cyclic invariance of rbg and considering indices modulo n + 1.Next denote by M [ i , . . . , i k ] the submatrix of M obtained by restricting to rows andcolumns i , . . . , i k . We see that L M f ( x i , . . . , x i k ) = L M [ i ,...,i k ] f ( x i , . . . , x i k ).Now, if { M [ i, i + 1] | M ∈ S } = I , then ( Q M ∈ S L M ) r ( x i , x i +1 ) = 0. Otherwise, wemust have M [ i, i + 1] = (cid:0) (cid:1) for all M ∈ S , or M [ i, i + 1] = (cid:0) − − (cid:1) for all M ∈ S .In the first case, we must have all M ∈ I n +1 with M [ i, i + 1] = (cid:0) − − (cid:1) contained in I n +1 \ S . The second case is similar. In either case, this implies that { M [ i, i + 2 , . . . , i + n ] | M ∈ I n +1 \ S } = { M [ i + 1 , . . . , i + n ] | I n +1 \ S } = I n and so( Y M ∈ I n +1 \ S L M ) q ( x i , x i +2 , . . . , x i + n ) = ( Y M ∈ I n +1 \ S L M ) q ( x i +1 , x i +2 , . . . , x i + n ) = 0 hus D n +1 { r, q } =0. (cid:3) Remark 8.3. Note that, in sufficiently high degree, r ( x , . . . , x n ) ∈ kerD n is equivalentto r ( x , . . . , x n ) ∈ P M ∈ I n ker L M . This second condition clearly holds for n = 2, andcan easily be shown to be preserved by the Ihara bracket. Hence, we can equivalentlystate Theorem 8.2 as the following: r ( x , . . . , x n ) ∈ X M ∈ I n ker L M for all r ( x , . . . , x n ) ∈ rbg . Remark 8.4. We have shown that, in block degree 1, bg is isomorphic as a vectorspace to the bigraded vector space of homogeneous polynomials satisfying Theorem 5.3,Example 6.3, Lemma 6.4, and whose reduced forms satisfy Corollary 7.4 and Theorem8.2. Note also that, as all these properties are preserved by the Ihara bracket, bg is a Liesubalgebra of the Lie algebra of homogeneous polynomials satisfying these properties.However, in block degree b , and weight w , we can only show that the dimension ofthe bigraded piece of the vector space of homogeneous polynomials satisfying theseconstraints is bounded above by Cw b − for some constant C .9. Deriving the Ihara action formula For elements of the double shuffle Lie algebra, the (linearised) Ihara action is givenby the following [6]: Proposition 9.1. For σ ∈ Lie [ e , e ] , u ∈ { e , e } × , the linearised Ihara action is givenrecursively by (9.1) σ ◦ e n e u := e n σe u − e n e σ ∗ u + e n e ( σ ◦ u ) where ( a . . . a n ) ∗ := ( − n a n . . . a . Translating the linearised Ihara action into the language of commutative variables,we find the following. Theorem 9.2. Let f ( x , . . . , x m ) be the image of the block degree m − part of σ ∈ Lie [ e , e ] , and g ∈ Q [ x , . . . , x n ] . Then the linearised Ihara action is given by ( f ◦ g )( x , . . . , x m + n − ) = n X i =1 ( − ( m +1)( i − f ( x i , x i +1 , . . . , x i + m − ) x i − x i + m − × (cid:18) (1 + ( − m +1 x i + m − x i ) g (¯ x , . . . , ¯ x i , x i + m , . . . , x m + n − ) − (1 + ( − m +1 x i x i + m − ) g (¯ x , . . . , ¯ x i − , x i + m − , . . . , x m + n − ) (cid:19) where we define ¯ x i := ( − m +1 x i .Proof. We start by writing, for u = u . . . u n ∈ { e , e } × , σ ◦ u . . . u n = ǫ σu . . . u n + n X i =1 ǫ i u . . . u i σu i +1 . . . u n here ǫ i ∈ { , ± } for each i . We first claim that ǫ i = 0 if u i = u i +1 . We take here u = e and u n +1 = e .If u i = u i +1 = e , then our recursive formula (9.1) shows ǫ i = 0, as σ does not ‘insert’between adjacent e . If u i = u i +1 = e , then our recursion gives us terms of the form · · · + u . . . u i − e σ ∗ e u i +2 . . . u n + u . . . u i − e σe u i +2 . . . u n + u . . . u i − e e σ ∗ u i +2 . . . u n + · · · As, for σ ∈ Lie[ e , e ], σ + σ ∗ = 0, the terms corresponding to u . . . u i σu i +1 . . . u n cancel, giving us that ǫ i = 0. Hence, our block-polynomial formula will consist of a sumover the blocks of u , each corresponding to the insertion of σ into a single block.We will induct on the number of blocks in u . If u consists of a single block, u = ( e e ) k ,and σ ◦ u = σ ( e e ) k + e σ ∗ e ( e e ) k − + e e σ ( e e ) k − + e e e σ ∗ e ( e e ) k − + . . . = k X i =0 [( e e ) i σ ( e e ) k − i + ( e e ) i e σ ∗ e ( e e ) k − − i ] . Letting f ( x , . . . , x m ) be the polynomial representing the block degree n part of σ and g ( x ) = x k +21 be the polynomial representing u , this is equivalent to the statementthat( f ◦ g )( x , . . . , x m ) = k X i =0 ( x x m ) i f ( x , . . . , x m ) g ( x m ) x m + ( − m +1 x x m k − X i =0 ( x x m ) i f ( x , . . . , x m ) g ( m ) x m = f ( x , . . . , x m ) x − x m (cid:18) ( x x m ) k +2 − − m +1 ( x x m ) k +1 − ( − m +1 x x m (cid:19) g ( x m )= f ( x , . . . , x m ) x − x m (cid:18) g ( x ) − g ( x m ) + ( − m +1 x m x g ( x ) − ( − m +1 x x m g ( x m ) (cid:19) , which is precisely the result given by the formula.Now suppose our formula is correct for words consisting of n − e ue be a word consisting of n blocks, i.e. e ue = b . . . b n , represented by the monomial g ( x , . . . , x n ). As we have merely appended a block onto the end of a word, the first n − f ◦ g ) will be given by our formula, by our induction hypothesis. Tosee this, consider ( f ◦ g alt ), where g alt is the polynomial corresponding to the word e u alt e = b . . . b ′ n − . Here b ′ n − is the smallest block extending b n − and ending on e .The Ihara action of any σ ∈ Lie[ e , e ] on u and u alt will produce terms that are identicalupon swapping b n − b n ↔ b ′ n − up to those terms in which σ inserts into b n − b n . Indeed,they will agree under this swapping until we consider terms in which σ inserts beyondthe end of b n − . Thus, it suffices to show that the formula holds for a word e ue = b b of block degree 1. e have 2 cases: the repeated letter in e ue is e , or it is e . In the first case, u = ( e e ) k e ( e e ) l and σ ◦ u = k X i =0 ( e e ) i σ ( e e ) k − i e ( e e ) l + k − X i =0 ( e e ) i e σ ∗ e ( e e ) k − − i e ( e e ) l + l X i =0 ( e e ) k e ( e e ) i σ ( e e ) l − i + l − X i =0 ( e e ) k e ( e e ) i e σ ∗ e ( e e ) l − − i In terms of commutative polynomials, after summing the geometric series, we obtain( f ◦ g )( x , . . . , x m +1 ) = f ( x , . . . , x m ) x − x m (cid:18) ( x x m ) k − − m +1 ( x x m ) k +1 − ( − m +1 x x m (cid:19) g ( x m , x m +1 )+ f ( x , . . . , x m +1 ) x − x m +1 (cid:18) ( x x m +1 ) l +2 − − m +1 ( x x m +1 ) l +1 − ( − m +1 x x m +1 (cid:19) g ( x , x m +1 ) . Simplifying, and noting that g ( x , x ) = x k +11 x l +22 , we obtain( f ◦ g )( x , . . . , x m +1 ) = f ( x , . . . , x m ) x − x m (cid:18) x m x g ( x , x m + 1) − g ( x m , x m + 1)+( − m +1 g ( x , x m +1 ) − ( − m +1 x x m g ( x m , x m +1 ) (cid:19) + f ( x , . . . , x m +1 ) x − x m +1 (cid:18) g ( x , x ) − g ( x , x m +1 )+( − m +1 x m +1 x g ( x , x ) − ( − m +1 x x m +1 g ( x , x m +1 ) (cid:19) . Considering parity, and defining ¯ x i := ( − m +1 x i , we can rewrite this as( f ◦ g )( x , . . . , x m +1 ) = ( − (0)( m +1) f ( x , . . . , x m ) x − x m (cid:18) g (¯ x , x m +1 ) + ( − m +1 x m x g (¯ x , x m +1 ) − g ( x m , x m +1 ) + ( − m +1 x m x m +1 g ( x m , x m +1 ) (cid:19) + ( − m +1 f ( x , . . . , x m +1 ) x − x m +1 (cid:18) g (¯ x , ¯ x ) − ( − m +1 x m +1 x g (¯ x , ¯ x ) − g (¯ x , x m +1 ) + ( − m +1 x x m +1 g (¯ x , x m +1 ) (cid:19) giving the desired formula. The second case follows similarly. ence, our general formula is ( f ◦ g )( x , . . . , x m + n − ) = A ± B ± C , where A = n − X i =1 ( − ( m +1)( i − f ( x i , x i +1 , . . . , x i + m − ) x i − x i + m − × (cid:18) (1 + ( − m +1 x i + m − x i ) g (¯ x , . . . , ¯ x i , x i + m , . . . , x m + n − ) − (1 + ( − m +1 x i x i + m − ) g (¯ x , . . . , ¯ x i − , x i + m − , . . . , x m + n − ) (cid:19) ,B = ( − ( m +1)( n − f ( x n − , . . . , x m + n − ) x n − − x n + m − × (cid:18) (1 + ( − m +1 x n + m − x n − ) g (¯ x , . . . , ¯ x n − , x m + n − ) − (1 + ( − m +1 x n − x n + m − ) g (¯ x , . . . , ¯ x n − , x n + m − , x m + n − ) (cid:19) ,C = ( − ( m +1)( n − f ( x n , . . . , x m + n − ) x n − x ” m + n − × (cid:18) (1 + ( − m +1 x n + m − x n ) g (¯ x , . . . , ¯ x n ) − (1 + ( − m +1 x n x n + m − ) g (¯ x , . . . , ¯ x n − , x m + n − ) (cid:19) , and the signs of B and C agree. To fix this sign, we need only to consider thesign of the term corresponding to ( x n − x m + n − ) f ( x n − ,...,x m + n − ) g ( x ,...,x n − ,x m + n − ,x m + n − ) x m + n − .This corresponds to inserting σ after the first two letters of the ( n − th block. Thesign will be positive if this block starts with an e and must have the same sign as( − m +1 otherwise. Let g ( x , . . . , x n ) = x d . . . x d n n . Then by Lemma 2.10 the ( n − th block starts with e if d + . . . + d n − ≡ n − e otherwise. Thus, thesign of the term corresponding to ( x n − x m + n − ) f ( x n − ,...,x m + n − ) g ( x ,...,x n − ,x m + n − ,x m + n − ) x m + n − is( − ( m − d + ... + d n − − n +2) . Comparing this with our formula, we see that the finaltwo terms must appear with a positive sign, giving the desired result. (cid:3) To obtain (4.2), we must translate this across into the ‘depth-signed’ convention.Specifically, we must find the action of the map e 7→ − e in terms of commutativevariables. Lemma 9.3. The automorphism Q h e , e i → Q h e , e i given by e 7→ − e , is equivalentunder the isomorphism (4.1) to the map Q [ x , . . . , x n ] → Q [ x , . . . , x n ] f ( x , . . . , x n ) ( − ⌈ l ⌉ f ( − x , x , . . . , ( − n x n )(9.2) for f a homogeneous polynomial of degree l + 2 . roof. Note that it suffices to show that, for a word w of length l and depth d , with π bl ( w ) = x d . . . x d n n , this congruence holds d ≡ ⌈ l ⌉ + d + d + · · · (mod 2) . We will induct on the number of blocks in e we . If e we consists of a single block,then e we = ( e e ) l +1 , and so d = l , and d = l + 2. Thus the result holds.Suppose the result holds for w such that e we consists of n blocks. Let e we = w ′ w d n +1 be a word of length l + 2 and depth d + 1,consisting of n + 1 blocks, where w d n +1 is a single block of length d n +1 and w ′ is a word of length l ′ and depth d ′ . Suppose π bl ( w ) = x d . . . x d n n x d n +1 n +1 .If l ′ is even, then w ′ = e ue consists of n ≡ d n +1 must be odd.So, by induction, d ′ − ≡ l ′ − 22 + X ≤ i +1 ≤ n d i +1 (mod 2) . Thus d = d ′ + ⌈ d n +1 ⌉ − ≡ l ′ − 22 + X ≤ i +1 ≤ n d i +1 (mod 2) + ⌈ d n +1 ⌉ (mod 2) ≡ ⌈ l ′ + d n +1 − ⌉ + X ≤ i +1 ≤ n +1 d i +1 (mod 2) ≡ ⌈ l ⌉ + X ≤ i +1 ≤ n +1 d i +1 (mod 2) , and so the result holds. Similar considerations for l ′ odd prove the result in general. (cid:3) Applying this transformation, and simplifying, we obtain Proposition 4.5, giving theformula( f ◦ g )( x , . . . , x m + n − ) = n X i =1 f ( x i , . . . , x i + m − ) x i − x i + m − (cid:18) x i g ( x , . . . , x i , x i + m , . . . , x m + n − ) − x i + m − g ( x , . . . , x i − , x i + m − , . . . , x m + n − ) (cid:19) . Further Remarks As bg is a free Lie algebra, we have a non-canonical isomorphism g m ∼ = bg , and hence,we should be able to lift relations such as cyclic insertion and the block shuffle relationsto g m . One would hope that we could follow the example of Brown in [5], in which he liftssolutions from ls to dmr , but it is as yet unclear how this should work. Progress on thishas been made by Hirose and Sato [12], who provide an ungraded version of the blockshuffle relations, but it is an area ripe for further consideration. Then, as the collection ofrelations presented in this paper, alongside block graded double shuffle relations, shouldcompletely describe all relations among block graded motivic multiple zeta values, a uccessful lifting of these relations could provide a complete description of all relationsamong motivic multiple zeta values, and give an approach to tackling questions aboutthe completeness of the associator and double shuffle relations. The construction ofgenuine relations, and the connection between these block graded relations and knownrelations is explored further in the author’s doctoral thesis. References [1] Schneps L. Baumard S. On the derivation representation of the fundamental Lie algebra of mixedelliptic motives. arXiv:1510.05549 , 2015.[2] F. Brown. Letter to Charlton (11/10/2016).[3] F. Brown. Mixed Tate motives over Z . Annals of Math. , 175(2):949 – 976, 2012.[4] F. Brown. Depth-graded motivic multiple zeta values. arXiv:1301.3053 , 2013.[5] F. Brown. Anatomy of an Associator. arXiv:1709.02765 , 2017.[6] F Brown. Zeta elements in depth 3 and the fundamental Lie algebra of a punctured elliptic curve. Forum Math. Sigma , 5:1–56, 2017.[7] S. Charlton. The alternating block decompositions of iterated integrals, and cyclic insertion onmultiple zeta values. arXiv:1703.03784 , 2017.[8] A. Deligne, P. Goncharov. Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. ´Ecole Norm.Sup. , 38:1–56, 2005.[9] V.G. 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