Depth-graded motivic multiple zeta values

We study the depth filtration on multiple zeta values, on the motivic Galois group of mixed Tate motives over $\mathbb {Z}$ and on the Grothendieck–Teichmüller group, and its relation to modular forms. Using period polynomials for cusp forms for $\mathrm {SL} _2(\mathbb {Z})$, we construct an explicit Lie algebra of solutions to the linearized double shuffle equations, which gives a conjectural description of all identities between multiple zeta values modulo $\zeta (2)$ and modulo lower depth. We formulate a single conjecture about the homology of this Lie algebra which implies conjectures due to Broadhurst and Kreimer, Racinet, Zagier, and Drinfeld on the structure of multiple zeta values and on the Grothendieck–Teichmüller Lie algebra.


Introduction
We begin by motivating the results of this paper from two apparently different, but in fact equivalent, perspectives.

Depth filtration on multiple zeta values
Multiple zeta values are defined for integers n 1 , . . . , n r−1 ≥ 1 and n r ≥ 2 by ζ(n 1 , . . . , n r ) = Their weight is the quantity n 1 + · · · + n r , and their depth is the number of indices r. Relations between multiple zeta values of depth 2 were first studied by Euler. Let Z N denote the Q-vector space spanned by multiple zeta values in weight N . Zagier conjectured, firstly, that the space Z of multiple zeta values is isomorphic to the direct sum of the Z N (in other words, the weight is a grading), and secondly, that the dimension of Z N can be expressed using the generating series (1.1)

F. Brown
right-hand side of (1.1). Furthermore, if one replaces Z N with the Q-vector space of motivic multiple zeta values ζ m (n 1 , . . . , n r ) of weight N , then (1.1) is a theorem [Bro12,Del13]. The rational function on the right-hand side of (1.1) can be interpreted as follows: it is the Poincaré series of the free graded module, generated by ζ m (2n) for n ≥ 1, over the graded dual of the universal enveloping algebra of the Lie algebra of the category of mixed Tate motives over Z, which is free with one generator in every odd degree ≤ −3 (see [DG05,Del13] for further details). Based on numerical experiments, Broadhurst and Kreimer [BK97] formulated a fascinating and more refined conjecture which takes into account the depth. The depth, by contrast with the conjectural properties of the weight, is only a filtration, and not a grading. Let Z N,d denote the Q-vector space spanned by multiple zeta values in weight N and depth d, modulo multiple zeta values of weight N and strictly lower depth. They propose that (1.2) where, using the notation from [IKZ06, Appendix], . (1.3) Formula (1.2) specializes to the statement (1.1) upon setting t = 1. The meaning of this conjecture is still mysterious, but one goal of this paper is to offer a homological interpretation of (1.2). The series E(s) and O(s) are the generating series for the dimensions of the spaces of even and odd single zeta values respectively, and S(s) is the generating series for the dimensions of the space of cusp forms for the full modular group SL 2 (Z). The first prediction of (1.2), due to the presence of a non-trivial coefficient of t 2 in the denominator of the right-hand side, is the existence of an extra relation between double zeta values of even weight for every cusp form, modulo multiple zeta values of lower depth (single zeta values). These relations have indeed been shown to exist and are well understood by the work of Gangl et al. [GKZ06], who exhibited an infinite family of such relations. The smallest one corresponds to the Ramanujan cusp form of weight 12: 28 ζ(3, 9) + 150 ζ(5, 7) + 168 ζ(7, 5) = 5197 691 ζ(12). (1.4) The coefficients in this and all such equations can be related to period polynomials for cusp forms, or equivalently, to group cocycles for SL 2 (Z). Furthermore, a geometric mechanism for these relations is by now fairly well understood [Bro14b]. The situation in higher depths remains very unclear. It is known by work of Zagier [Zag93] and Goncharov [Gon01b] that (1.2) is true (in a suitable setting, i.e. for solutions to the double shuffle equations as discussed below) in depths 2 and 3, respectively. Nevertheless, the presence of the term in t 4 in the right-hand side of (1.2) suggests a new phenomenon in depth 4. If we interpret the right-hand side of (1.2) in terms of the Poincaré series of a depth-graded version of the Lie algebra of the category of mixed Tate motives over Z, the term in t 4 suggests the existence of new generators in this Lie algebra in depth 4, corresponding to cusp forms for the full modular group.
In this paper we supply candidates for these 'exceptional' generators by constructing them explicitly out of period polynomials of cusp forms. As a result, we can formulate a much more precise conjecture than (1.2) which predicts not only the dimensions of the spaces of multiple 530 Depth-graded motivic multiple zeta values zeta values in all weights and depths, but also their relations and modulo lower depths. In order to get some sense of these exceptional generators, consider the first one, which occurs in depth 4 and weight 12. It turns out that all multiple zeta values in weight 12 and depth 4 are proportional to a single element ζ(1, 1, 2, 8) modulo terms of lower depth and products, for example, ζ(4, 3, 3, 2) ≡ 116 ζ (1, 1, 2, 8).
The exceptional generator corresponding to the Ramanujan cusp form Δ annihilates every such relation, and therefore gives an interpretation of the coefficients (in this case, the number 116) in terms of ratios of critical values of the L-function of Δ.

The projective line minus three points
Let G dR MT (Z) = Aut ⊗ MT (Z) (ω dR ) denote the motivic Galois group of the Tannakian category MT (Z) of mixed Tate motives over the integers [DG05], and let U dR MT (Z) ≤ G dR MT (Z) denote its unipotent radical. One has a canonical isomorphism G dR G m . Let 0 Π 0 = π dR 1 (P 1 \{0, 1, ∞}, → 1 0 ) and 0 Π 1 = π dR 1 (P 1 \{0, 1, ∞}, denote the de Rham fundamental group (respectively, torsor of paths) at a tangential base point at 0 (respectively, between tangential basepoints at 0 and 1) [Del89]. The latter is a torsor over the former, via composition 1 of paths 0 Π 0 × 0 Π 1 → 0 Π 1 . Since 0 Π 0 , 0 Π 1 are the de Rham realizations of (pro)-objects in the category of mixed Tate motives over Z, the group G dR acts upon them, and there is a canonical representation where the group of automorphisms on the right denotes those automorphisms which respect the structure 0 Π 0 × 0 Π x → 0 Π x for x ∈ {0, 1}. This action is the motivic version of the outer action of the absolute Galois group Gal(Q/Q) on the pro-completion of the fundamental group of X which was first studied extensively by Deligne, Drinfeld, and Ihara [Del13,Dri90,Iha02]. Deligne conjectured that this action is faithful, or equivalently, that the motivic torsor of paths on X generates the category MT (Z), which was shown in [Bro12,Del13]. The graded Lie algebra 2 of 0 Π 0 is the free graded Lie algebra L(e 0 , e 1 ) on two generators e 0 , e 1 . Therefore, the infinitesimal version of the action of U dR MT (Z) on 0 Π 0 gives a very concrete way to study the motivic Galois group, or equivalently, its graded Lie algebra g m together with its representation: H 1 (g m ; Q) = (g m ) ab , one can rephrase the previous theorem by saying that (1.5) The abstract structure of the Lie algebra g m is therefore very simple, but the information about linear relations between multiple zeta values is encoded in the coefficients of the generators σ 2n+1 , which are not known explicitly. In this paper we study the associated graded version dg m of this Lie algebra for a filtration called the depth, which is related to the depth filtration on multiple zeta values. By contrast with the case of g m , we will provide an explicit conjectural description of all the generators of dg m . It is not a free Lie algebra, but a little more complicated.

Depth filtration on the motivic Lie algebra
As defined in [DG05], the depth filtration is induced geometrically by the inclusion P 1 \{0, 1, ∞} ⊂ G m and is the decreasing filtration D on L(e 0 , e 1 ), where D r consists of Lie brackets containing at least r occurrences of the letter e 1 . It is preserved by the Ihara bracket { , }, and therefore defines a filtration on g. One defines the depth filtration D r g m to be the induced filtration. An element in g m lies in D r g m if and only if the corresponding derivation of Lie 0 Π 0 = L(e 0 , e 1 ) sends Depth-graded motivic multiple zeta values D s to D s+r for all s. This makes the motivic nature of D clear since the σ 2n+1 increase the depth by at least 1.
We consider the associated graded Lie algebra: It is bigraded by weight and depth: the depth will be indicated by a subscript, thus dg m r = gr r D g m . We identify dg m with a Lie subalgebra of gr D g which as a bigraded vector space is identified with L(e 0 , e 1 ). Note that, in fact, the Ihara bracket { , } respects the depth grading on the free Lie algebra L(e 0 , e 1 ), so the Lie bracket on gr D g is given by the identical formula to that for { , }.
In depth 1, dg m inherits canonical 'zeta' generators which are the images of σ 2n+1 : (1.6) Ihara discovered, astonishingly, that in the depth-graded Lie algebra the generatorsσ 2n+1 are not free. The first relation occurs in weight 12: In order to reconcile this relation with the freeness theorem (Theorem 1.1), there must exist an extra generator in dg m in weight 12 to compensate for it: the new generator is given by the lowest depth part of {σ 3 , σ 9 } − 3{σ 5 , σ 7 }, which is in depth ≥ 4. Exceptional generators such as these (but defined in a direct and somewhat different manner) are one of the main objects of study of this paper. The general quadratic relations between theσ 2n+1 have been known explicitly for some time [IT93, Sch06, Gon05, GKZ06] and will be re-derived as an immediate consequence of the formalism we introduce below. To describe them, let V n = i+j=n QX i Y j denote the vector space of homogeneous polynomials of degree n in two variables, with its right action of SL 2 (Z). Evaluating cocycles on the matrix 0 −1 1 0 induces an isomorphism where + denotes invariants under the involution which sends a cocycle γ → C γ to the cocycle γ → C γ −1 , where = 1 0 0 −1 . One can show that this involution is induced by the action of complex conjugation on (C\R)/GL 2 (Z). The space S 2n ⊂ Q[X, Y ] is the space of even-period polynomials: it is the space of antisymmetric homogeneous polynomials P (X, Y ) of degree 2n − 2, divisible by Y , satisfying P (±X, ±Y ) = P (X, Y ) and One shows that the quadratic relations betweenσ 2i+1 in weight 2n + 2 are completely described by period polynomials in S 2n : (1.8) where i + j = n in both equations. The first goal of this paper is to provide a complete conjectural description of dg m . 533 F. Brown

Results
1.4.1 Missing generators. Firstly, we construct the candidate exceptional generators in depth 4 using period polynomials for cusp forms. We define an explicit map e : H 1 cusp (SL 2 (Z); V 2n−2 ) + −→ D 4 g which, to every even-period polynomial associates a Lie word in two generators e 0 , e 1 , of degree 4 in e 1 . A different source of extra generators in depth 4 are the lowest-depth part of expressions of the form i<j λ i,j {σ 2i+1 , σ 2j+1 } where the λ i,j are as in (1.8) (these expressions have depth ≥ 4) but such generators depend on a choice of generators σ 2n+1 . A canonical choice of such generators was given in [Bro17a], but the relationship with the elements e, which are introduced in the present paper and defined very differently, is not completely understood. For instance, even if one works modulo commutators, these two possible definitions of generators in depth 4 are related by a non-trivial isomorphism on the space of period polynomials (see § § 1.4.3, 8.4). The simplest possible conjecture that one can make is that the depth-graded motivic Lie algebra is generated by the canonical generatorsσ 2n+1 in depth 1, the image of the exceptional map e in depth 4 and subject only to the known quadratic relations between theσ 2n+1 in depth 2. This is equivalent to a statement about H i (dg m ; Q) for i = 1, 2, and suggests the following reformulation of the Broadhurst-Kreimer conjecture.
Conjecture 1. The image of e lies in dg m , and (1.9) We show in § 10.2 that Conjecture 1 implies the version of (1.2) in which multiple zeta values are replaced by motivic multiple zeta values. This is in turn equivalent to (1.2) if one assumes the period conjecture for mixed Tate motives over Z.
Conjecture 1 describes all relations between depth-graded motivic multiple zeta values (modulo products and ζ m (2)). More precisely, consider the ring H of shuffle-regularized motivic multiple zeta values ζ m (w) where w is a word in {e 0 , e 1 }. It is weight-graded, and also has a depth filtration D, which counts the number of e 1 's. Let gr D H denote the associated graded ring and let gr D I denote its augmentation ideal. Then a linear relation of weight N and depth d of the form where λ w ∈ Q and w ranges over words in e 0 , e 1 of length N with d letters e 1 , holds in the quotient 4 (gr D H)/((gr D I) 2 + (ζ m (2))) if and only if, for all x ∈ dg m d of weight N and depth d of the form x = w c w w, one has w c w λ w = 0. Since Conjecture 1 provides an explicit presentation for dg m , it conjecturally describes all such relations between (motivic) depth-graded multiple zeta values (see Example 8.4).
In fact, the formalism developed here enables one to describe all relations between depthgraded motivic multiple zeta values (and not necessarily modulo products or modulo ζ m (2)). These are described by the linear forms which vanish not on dg m but rather on the bigraded right-module over the universal enveloping algebra Udg m generated by the depth 1 components of a rational associator τ (1) (defined in [Bro17a]). This can be made completely explicit: see § 6.7.
1.4.2 Linearized double shuffle equations. The main evidence for the previous conjecture comes from the double shuffle equations. It is known that where grt is Drinfeld's Grothendieck-Teichmüller Lie algebra, and dm 0 is Racinet's regularized double shuffle Lie algebra, both of which are defined by explicit equations. The inclusion of g m in grt and dm 0 follow from Theorem 1.1 and results of Drinfeld [Dri90] and Racinet [Rac02], respectively. The inclusion grt ⊂ dm 0 is due to Furusho [Fur11].
If we pass to the depth-graded Lie algebras, we have where ls are the linearized double shuffle equations defined in [IKZ06]. The advantage of these equations are that they are extremely simple to define: ls is essentially the intersection of two shuffle algebras. Hitherto, ls was studied merely as a vector space, but it turns out, as a consequence of the work of Racinet, that it also inherits a Lie algebra structure for the linearized Ihara bracket. We offer a complete conjectural description of ls below. The first theorem states that the exceptional elements are solutions to the linearized double shuffle equations in depth 4. (1.10) The formula for the map e is given in § 8, and associates to every even-period polynomial f a solution e f of the linearized double shuffle equations. The question of whether the elements e f are motivic (i.e. whether they lie in the subspace dg m ) is open.  This conjecture states, in particular, that ls is generated by zeta elements and the exceptional generators in depth 4 subject only to the known quadratic relations between zeta elements. It is at the same time, the simplest and the strongest conjecture that one can formulate. 535

F. Brown
It implies several open conjectures about relations between multiple zeta values. For example, it implies: Conjecture 1, and hence the motivic version of the Broadhurst-Kreimer conjecture; the conjecture dg m = ls (which in turn implies a conjecture in [IKZ06]); and the conjectures g m = grt (Drinfeld) and g m = dm 0 (Zagier, Racinet). The proofs of these implications use Theorem 1.1 and Furusho's theorem [Fur11] in an essential way. Since the Lie algebra ls is defined in a very simple and completely elementary way, the previous conjecture suggest a possibility of seeking a proof of all of the above conjectures intrinsically within the theory of modular forms.
1.4.3 Discussion. Note that the vanishing of H i (ls) for i ≥ 3 is equivalent to the vanishing for i = 3. This follows from the well-known fact that the vanishing of a Yoneda Ext group causes all higher Ext groups to vanish ( [EL16], [Bro17a,Remark 8.6]).
The exceptional elements e satisfy a range of special properties which are studied in § § 8 and 9. Using these, we can prove that they satisfy no quadratic relations, which provides some meagre evidence in favour of the previous conjectures. We do not know, however, that the e f are non-trivial in the abelianization of ls, nor can we presently rule out the existence of relations among the elements of the form which are in depth 5 and in weights ≥ 15 (respectively, 17). Relations which are quadratic in the e f could first occur in weight 28 and depth 8. Viewed from this perspective, the current numerical data in favour of the standard conjectures on multiple zeta values is lacking, since new phenomena could potentially occur in weights and depths beyond the range of current experimentation. Any such phenomena would point to new and fascinating connections between mixed Tate motives over Z with geometry and arithmetic. Indeed, the methods introduced in this paper should enable one to test the validity of Conjecture 2 to far higher weights than presently known.
The motivic Lie algebra g m together with its depth filtration D naturally gives rise to a spectral sequence ( § 4.5), and the Broadhurst-Kreimer conjecture is equivalent to the statement that this spectral sequence should behave as simply as possible (given the existence of the quadratic relations betweenσ 2n+1 ); there is only one non-trivial differential, This is possibly also an argument in favour of Conjecture 1. In [Bro17a] we computed this differential by finding canonical lifts of the zeta elements to depth 3. The relation with the map e is mysterious, although Yasuda has subsequently found a conjectural relation between e and the image of d in the abelianization of ls (private correspondence) involving critical values of L-functions of cusp forms. See Example 8.5. Finally, we investigate the Lie subalgebra of dg m which is generated only by the elements σ 2n+1 (without exceptional elements), and conjecture that it describes the structure of totally odd depth-graded motivic multiple zeta values.

Contents of the paper
In § § 2-4 we recall some background on the motivic fundamental group of P 1 \{0, 1, ∞}, the Ihara action and the depth filtration. In § 5 we discuss the linearized double shuffle relations from the Hopf algebra point of view. In § 6, and throughout the rest of the paper, we use polynomial 536 Depth-graded motivic multiple zeta values representations to replace words of fixed D-degree r in e 0 , e 1 with polynomials in r variables: which sends words beginning in e 0 to zero, andρ(e 1 e n 1 0 . . . e 1 e nr 0 ) = x n 1 1 . . . x nr r . This replaces identities between non-commutative formal power series with functional equations in commutative polynomials, strongly reminiscent of those considered in [Eca03, IKZ06]. We show that in the polynomial representation, the Ihara bracket has an extremely simple form ( § 6.5). A simple way to view the duality between depth-graded motivic multiple zeta values and polynomials is via the generating series The generators (1.6) are simply the coefficients of ζ m D (2n + 1): In § 7 we review the relation between period polynomials and depth 2 multiple zeta values, and in § 8 we define, for each period polynomial P , an element which defines the exceptional elements in ls 4 . These elements satisfy some remarkable properties ( § 9) which are stable under the Ihara bracket. In § 10 we discuss Conjecture 1 and its consequences, and in § 11 we discuss some applications for the enumeration of the totally odd multiple zeta values ζ(2n 1 + 1, . . . , 2n r + 1) where n i ≥ 1.
1.5.1 Related work. Since the first draft of this paper appeared, there have been a number of related developments, which cannot all be mentioned here for reasons of space. I apologize to the many people who have subsequently extended some of the ideas in this paper in various directions [Mat16, Tas16, Ma16, Li19], not only for the delay in publishing this work, but also for being unable to give a complete survey of subsequent developments of the subject here.
First of all, an alternative description of depth 4 exceptional generators was given in [Bro17a], by constructing canonical zeta elements in g m /D 4 g m and applying the differential in the depth spectral sequence. These elements are motivic (i.e. lie in dg m 4 ), but their relationship to the exceptional elements e defined here is far from clear. The entire construction relies on the relationship between P 1 \{0, 1, ∞} (genus 0) and the unipotent completion of the fundamental group Tate elliptic curve (genus 1), and the fact that the depth filtration is induced by natural filtrations in the elliptic setting. Similarly, the precise relationship between the work of Pollack [Pol09] and the present paper is now partially understood but warrants further investigation.
Subsequently, the work [Bro14b] explains the origin of the quadratic relations between σ 2n+1 by proving that they come from 'modular elements' corresponding to non-critical values of L-functions of cusp forms, which act on the relative completion of the fundamental group on the moduli stack M 1,1 of elliptic curves. From this perspective, the canonical generators σ 2n+1 can be understood as coming from Eisenstein series, which supports the philosophy that Conjecture 2 relates entirely to modular forms. Nevertheless, the connection between modular elements and the phenomena in depth 4 studied in the present paper remain unexplored.

F. Brown
In a very different direction, Goncharov [Gon05] has studied the depth filtration from the perspective of the homology of the general linear group GL d (Z), which he relates to the structure of multiple zeta values in depth d. We have no understanding of the relationship between this point of view and conjecture (1.11).
After the first draft of this paper appeared, Conjecture 4 was related to questions of Koszulity in the paper [EL16].
Finally, the interested reader may wish to consult the notes 5 where, in answer to a question due to Zagier, we reformulated the Broadhurst-Kreimer conjecture as a short exact sequence, which suggests another line of attack on Conjecture 2.

Reminders on π m
1 (P 1 \{0, 1, ∞}) 2.1 The motivic π 1 of P 1 \{0, 1, ∞} Let X = P 1 \{0, 1, ∞}, and let → 1 0 , − → 1 1 denote the tangential base points on X given by the tangent vector 1 at 0, and the tangent vector −1 at 1. Denote the de Rham realization of the motivic fundamental torsor of paths on X with respect to these basepoints by It is the affine scheme over Q which to any commutative unitary Q-algebra R associates the set of group-like formal power series in two non-commuting variables e 0 and e 1 , where Δ is the completed coproduct for which the elements e 0 and e 1 are primitive: Δe i = 1 ⊗ e i + e i ⊗ 1 for i = 0, 1. The ring of regular functions on 0 Π 1 is the Q-algebra whose underlying vector space is spanned by the set of words w in the letters e 0 , e 1 , together with the empty word, and whose multiplication is given by the shuffle product x : Q e 0 , e 1 ⊗ Q Q e 0 , e 1 → Q e 0 , e 1 . The deconcatenation of words defines a coproduct, making O( 0 Π 1 ) into a Hopf algebra. This gives rise to a group structure on 0 Π 1 (Q) (corresponding to the fact that in the de Rham realization there is a canonical path between any two points on X). Any word w in e 0 , e 1 defines a function The Lie algebra of 0 Π 1 (Q) is the completed Lie algebra L(e 0 , e 1 ) ∧ of the graded Lie algebra L(e 0 , e 1 ) which is freely generated by the two elements e 0 , e 1 in degree −1. The universal enveloping algebra U L(e 0 , e 1 ) of L(e 0 , e 1 ) is the tensor (co)algebra on e 0 , e 1 : (2.1) It is the graded cocommutative Hopf algebra which is the graded dual of O( 0 Π 1 ). Its multiplication is given by the concatenation product, and its coproduct is the unique coproduct for which e 0 and e 1 are primitive. 5 www.ihes.fr/brown/BKExactSeq1.pdf.

538
Depth-graded motivic multiple zeta values

Action of the motivic Galois group
Now let MT (Z) denote the Tannakian category of mixed Tate motives over Z, with canonical fiber functor given by the de Rham realization. Let G MT (Z) denote the group of automorphisms of this fiber functor. It is an affine group scheme over Q. It has a decomposition as a semi-direct product where U MT (Z) is pro-unipotent. Furthermore, one knows from the relationship between the Ext groups in MT (Z) and Borel's results on the rational algebraic K-theory of Q that the graded Lie algebra of U MT (Z) is non-canonically isomorphic to the Lie algebra freely generated by one generator σ 2i+1 in degree −(2i + 1) for every i ≥ 1. It is important to note that only the classes of the elements σ 2i+1 in the abelianization U ab MT (Z) are canonical, the elements σ 2i+1 themselves are not.
Since O( 0 Π 1 ) is the de Rham realization of an Ind-object in the category MT (Z), there is an action of the motivic Galois group G MT (Z) on 0 Π 1 and hence (2. 2) The action of U MT (Z) on the unit element 1 ∈ 0 Π 1 defines a map and the action (2.2) factors through a map first computed by Ihara. It is obtained from [DG05, § § 5.9, 5.13], by reversing all words in order to be consistent with our convention for composition of paths. An element a ∈ 0 Π 1 defines an action denoted by a 0 on 0 Π 0 which is compatible with (2.4) via the composition of paths Therefore, writing x 00 (respectively, x 01 ) for the element x in 0 Π 0 (respectively, 0 Π 1 ), one finds that for g ∈ 0 Π 1 , where a 0 acts on the generators exp(e i ) in 0 Π 0 , for i = 0, 1, by We now give a concrete way to compute • by expressing it as the restriction of a combinatorially defined map • on a larger space. where a • e n 0 = e n 0 a, and for any a i ∈ {e 0 , e 1 }, (a 1 . . . a n ) * = (−1) n a n . . . a 1 .

F. Brown
We now show that the map • restricts to the full action induced by (2.4). To see this, identify the universal enveloping of the graded Lie algebra U Lie 0 Π 1 = UL(e 0 , e 1 ) with T (e 0 , e 1 ). Since g is isomorphic (as a vector space) to L(e 0 , e 1 ), there is also a natural embedding i : g → T (e 0 , e 1 ).
Proposition 2.2. The action (2.8) is obtained by restriction of • , that is, Identifying the (vector space) g ∼ = L(e 0 , e 1 ) with its image in T (e 0 , e 1 ), it follows that the infinitesimal, weight-graded, version of the map (2.6) is the derivation f 0 : L(e 0 , e 1 ) → L(e 0 , e 1 ) which for any f ∈ L(e 0 , e 1 ) is given by Adding the term e m 0 0 e 1 . . . e 1 e mr 0 f corresponding to concatenation on the right by f gives precisely the map defined by (2.7).

The motivic Lie algebra
By (2.3), we obtain a map of Lie algebras where the Ihara bracket satisfies {f, g} = f • g − g • f . It follows from Theorem 1.1 that this map is injective [Bro12]. In this paper, we shall identify Lie(U MT (Z) ) with its image, and abusively call it the motivic Lie algebra.
Definition 2.3. The motivic Lie algebra g m ⊆ g is the image of the map (2.9).
The Lie algebra g m is non-canonically isomorphic to the free Lie algebra with one generator σ 2i+1 in each degree −(2i + 1) for i ≥ 1.

Motivic multiple zeta values
Let A MT denote the graded Hopf algebra of functions on U MT (Z) . Dualizing (2.2) gives the motivic coaction (written in this paper as a left coaction) Furthermore, the image in 0 Π 1 (C) of the straight path dch from 0 to 1 in X under the comparison isomorphism is the Drinfeld associator element Φ ∈ 0 Π 1 (R) which begins

Depth-graded motivic multiple zeta values
The map which takes the coefficient of a word w in Φ defines the period homomorphism Here, we use the convention from [Bro12]: the coefficient of the word e a 1 . . . e an in Φ, for a i ∈ {0, 1}, is the iterated integral 1 0 ω a 1 . . . ω an regularized with respect to the tangent vector 1 at 0, and −1 at 1, where the integration begins on the left, and ω 0 = dt/t and ω 1 = dt/(1 − t).
Definition 3.1. The algebra of motivic multiple zeta values is defined as follows. The ideal of motivic relations between multiple zeta values is defined to be J MT ≤ O( 0 Π 1 ), the largest graded ideal contained in the kernel of per which is stable under Δ M . We set and let ζ m (n 1 , . . . , n r ) denote the image of the word e 1 (e 0 ) n 1 −1 . . . e 1 (e 0 ) nr−1 in H. Likewise, for any a 1 , . . . , a n ∈ {0, 1}, we let I m (0; a 1 , . . . , a n ; 1) denote the image of the word e a 1 . . . e an in H. Equivalently, one can define I m (0; a 1 , . . . , a n ; 1) = [O(π mot 1 (X, where the right-hand side is a motivic period [Bro14a] of the category MT (Z), and define the motivic multiple zeta values by specializing to the case when a 1 = 1 and a n = 0. The space H is naturally graded by the weight, and has a graded coaction and a period map per : H → R. The period of ζ m (n 1 , . . . , n r ) is ζ(n 1 , . . . , n r ). One obtains partial information about the motivic coaction (3.1) using the fact that it factors through the coaction which is dual to the Ihara action (2.4).
Since the motivic coaction on H factors through the Ihara coaction, it follows that the degree (r, ·) component factors through operators given by the same formula as (3.3) in which each term I is replaced by its image I m in H (respectively, A). Since A MT is cogenerated in odd degrees only, the motivic coaction on H is completely determined by the set of operators D 2r+1 for all r ≥ 1 (see [Bro12]).

Definition
The depth filtration was defined in [DG05, § 6.1]. We recall the definition in a slightly different language. The inclusion P 1 \{0, 1, ∞} → P 1 \{0, ∞} induces a map on the motivic fundamental groupoids π mot 1 (X, is nothing other than the deconcatenation coproduct, the filtration D d O( 0 Π 1 ) is given by with respect to which O( 0 Π 1 ) is a filtered comodule over the filtered Hopf algebra O( 0 Π 0 ), with respect to Δ dec . Furthermore, since the map (4.1) is motivic, it follows that D 0 is preserved by the action of the motivic Galois group. The same is true for all d by induction: if g ∈ G dR

MT (Z)
and g D i ⊂ D i for all i < d, then we also have gD d ⊂ D d by definition of D d , since g commutes with Δ dec , which is motivic. Therefore the depth is also motivic and descends to the algebra H.

542
Depth-graded motivic multiple zeta values By Definition 3.1 the depth filtration D d H is the increasing filtration defined by Following [Del10], it is convenient to define the D-degree on O( 0 Π 1 ) to be the degree in e 1 . It defines a grading on 0 Π 1 which is not motivic. By (4.2), the depth filtration (which is motivic) is simply the increasing filtration associated to the D-degree.

Depth-graded motivic multiple zeta values
where ζ m D (2) n are in depth 1 for all n ≥ 1.
Proof. Choose a homomorphism π 2 : H → Q[ζ m (2)] which respects the weight-grading and such that π 2 (ζ m (2)) = ζ m (2 By a motivic version [Bro12] of Euler's theorem, ζ m (2) n is a rational multiple of ζ m (2n), and so all powers of ζ m (2) have depth 1. The depth filtration thus satisfies Since D 0 H = Q, it follows trivially that π 2 respects the depth filtration because it is Q-linear.
Since the depth filtration is motivic, it is also respected by Δ M , and therefore the map (id ⊗ π 2 ) Δ M : H → A ⊗ Q Q[ζ m (2)] defined above also respects the depth filtration. The statement (4.3) follows since we know that this map is an isomorphism (by [Bro12, (2.13)] or [Del10, Proposition 5.8]).
Since gr D A is a commutative graded Hopf algebra (for the coproduct induced by Δ M ), it is a polynomial algebra. The same is then true for gr D H, by (4.3). If I denotes indecomposable elements, then it follows that I(gr D A) ∼ = gr D IA.

Depth-graded motivic Lie algebra
The depth filtration defines a decreasing filtration D r on g m where D r consists of words with at least r occurrences of e 1 : D r g m = w ∈ g m : deg e 1 w ≥ r .
We denote the associated graded Lie algebra by dg m . There is correspondingly a decreasing depth filtration on Ug m , also denoted by D.

F. Brown
It follows from the definitions above that dg m is the bigraded Lie algebra dual to the bigraded coalgebra I(gr D A) ∼ = gr D IA of depth-graded motivic multiple zeta values modulo products. Thus the problem of studying relations between (motivic) multiple zeta values modulo lower depth (and modulo ζ m (2)) and the algebra dg m are equivalent.

Depth-parity
The following proposition is a consequence of Tsumura's result on double shuffle equations (Proposition 6.4). Equivalently, if n 1 + · · · + n r ≡ r (mod 2), and n 1 + · · · + n r > 2 then ζ m D (n 1 , . . . , n r ) ≡ 0 (mod products). (4.4) We do not need to work modulo ζ m D (2) in the previous equation since the weight is larger than 2 by assumption and all other even zeta values ζ m (2n), n ≥ 2, are products.

Depth spectral sequence
The depth filtration on the motivic Lie algebra g m induces a homology spectral sequence which converges to the associated graded for the depth of the homology of g m . By Theorem 1.1, the latter satisfies and is entirely concentrated in depth 1. The depth spectral sequence satisfies where E 1 −p,q vanishes unless p ≥ 1 and p < q ≤ 2p, as can easily be seen from the Chevalley-Eilenberg chain complex which computes the homology of a Lie algebra (see § 10.2 and (10.3)). The differentials satisfy d r p,q : E r p,q → E r p−r,q+r−1 .
Proposition 4.4. The differentials d r vanish if r is odd.
Proof. The weight grading on dg m induces a weight grading on the depth spectral sequence, with respect to which the differentials are of degree 0. It follows from Proposition 4.3 and E 1 −p,q = gr p D H q−p (dg m ) that E 1 p,q , and hence all E r p,q , vanish unless the depth and weight have the same parity. The result follows since d r has (weight, depth)-bidegree (0, r). 544

Depth-graded motivic multiple zeta values
Here is a picture of the potentially non-vanishing E 1 = E 2 terms: . In fact, we know by Theorem 1.1 that and therefore everything to the left of the column indexed −1 converges to zero. Furthermore, one knows that gr 2 D H 1 (dg m ) = 0, and gr 3 D H i (dg m ) vanish for all i. One also has a complete description of gr 2 D H 2 (dg m ) in terms of period polynomials of cusp forms, as discussed below. Therefore, the first interesting part of the differential is and the main Conjecture 1 can be reformulated as saying that the components of all other differentials in the depth spectral sequence vanish.

Reminders on the standard relations
We briefly review the double shuffle relations and their depth-linearized versions. See [Car02,Rac02,Bro17b] for further background.
5.1.1 Shuffle product. Consider the algebra Q e 0 , e 1 of words in the two letters e 0 , e 1 , equipped with the shuffle product x ( § 2.1). It is defined recursively by for all w, w ∈ {e 0 , e 1 } × and i, j ∈ {0, 1}, and the property that the empty word 1 satisfies 1 x w = w x 1 = w. It is a Hopf algebra for the deconcatenation coproduct. A linear map Φ : Q e 0 , e 1 → Q is a homomorphism for the shuffle multiplication, or Φ w Φ w = Φ w x w for all w, w ∈ {e 0 , e 1 } × and Φ 1 = 1, if and only if the series One says that Φ satisfies the shuffle relations modulo products if either of the equivalent conditions (5.3) holds. The algebra Q e 0 , e 1 is bigraded for the degree, or weight (for which e 0 , e 1 both have degree −1), and the D-degree for which e 1 has degree 1, and e 0 degree 0. The relations (5.1)-(5.3) evidently respect both gradings. In this case, then, passing to the depth grading does not change the relations in any way, and the linearized shuffle relations are identical to the shuffle relations modulo products.
Equivalently, Φ w x w = 0 for all words w, w ∈ {e 0 , e 1 } × of total weight N and total D-degree r.

Stuffle product.
Let Y = {y n , n ≥ 1} denote an alphabet with one letter y i in every degree ≥ 1, and consider the graded algebra Q Y equipped with the stuffle product [Rac02]. It is defined recursively by y i w * y j w = y i (w * y j w ) + y j (y i w * w ) + y i+j (w * w ) (5.4) for all w, w ∈ Y × and i, j ≥ 1, and the property that the empty word 1 satisfies 1 * w = w * 1 = w. A linear map Φ : Q Y → Q is a homomorphism for the stuffle multiplication, or Φ w Φ w = Φ w * w for all w, w ∈ Y × and Φ 1 = 1, if and only if One says that Φ satisfies the stuffle relations modulo products if either of the equivalent conditions (5.7) holds. 546 Depth-graded motivic multiple zeta values The algebra Q Y is graded for the degree (where y n has degree n), and filtered for the depth (where y n has depth 1). By inspection of (5.4), we notice that the rightmost term is of lower depth than the other terms and therefore drops out in the associated graded. The associated graded of * therefore satisfies the same recursive definition as for x and it follows that the associated depth-graded is simply the shuffle algebra on Y . The depth induces a decreasing filtration on the (dual) completed Hopf algebra Q Y , and it follows from (5.5) that the images of the elements y n are primitive in the associated graded. Thus coproduct for which the elements y n are primitive, and which is a homomorphism for concatenation.
Definition 5.2. Let Φ ∈ T (Y ), the tensor (co)algebra on Y , of degree N ≥ 2 and D-degree r.
It defines a linear map w → Φ w on words in Y of weight N and D-degree r. We say that it satisfies the linearized stuffle relations if Racinet proved that dm 0 (Q) is preserved by the Ihara bracket. Since the latter is homogeneous for the D-degree, it follows that gr D dm 0 (Q) is also preserved by the bracket, but this is not quite enough to prove that ls is too.
Theorem 5.5. The bigraded vector space ls is preserved by the Ihara bracket.
Proof. The compatibility of the shuffle product with the Ihara bracket follows by definition. It therefore suffices to check that the linearized stuffle relation is preserved by the bracket. The proof of [Rac02] goes through identically, and uses in an essential way the fact that the images of the elements y 2n in gr 1 D dm 0 (Q) are zero (which in dm 0 (Q) follows from [Rac02, Proposition 2.2], but holds in ls by Definition 5.3).
Many thanks to a referee for pointing out at least two places in the literature which have subsequently provided a more detailed proof of this statement: [Maa19] and also [Sch15,Theorem 3.4.3]. It would be interesting to know if a suitable linearized version of the associator relations is equivalent to the linearized double shuffle relations.

Summary of definitions
We have the following Lie subalgebras of the Lie algebra g ∼ = (L(e 0 , e 1 ), { , }), equipped with the Ihara bracket: where g m is the image of the (weight-graded) Lie algebra of U MT (Z) and is isomorphic to the free graded Lie algebra on (non-canonical) generators σ 2i+1 for i ≥ 1. A standard conjecture states that g m ⊆ dm 0 (Q) is an isomorphism. Passing to the depth-graded versions, and writing dg m = gr D g m , we have inclusions of bigraded Lie algebras where ls stands for the linearized double shuffle algebra. Once again, all Lie algebras in (5.11) are conjectured to be equal. The bigraded dual space (dg m ) ∨ is isomorphic to the Lie coalgebra of depth-graded motivic multiple zeta values, modulo ζ m (2) and modulo products. The Lie algebras (5.11) are Lie subalgebras of gr D g ∼ = g, since the Ihara bracket is homogeneous with respect to D-degree.
All the above Lie algebras can, in particular, be viewed inside the vector space UL(e 0 , e 1 ) = T (Qe 0 ⊕ Qe 1 ), which is graded for the D-degree. Next, we show that complicated expressions in the non-commutative algebra T (Qe 0 ⊕ Qe 1 ) can be greatly simplified by encoding words of fixed D-degree in terms of polynomials. 548 Depth-graded motivic multiple zeta values 6. Polynomial representations

Composition of polynomials
Recall from (2.1) that U L(e 0 , e 1 ) is isomorphic to the bigraded tensor algebra T (e 0 , e 1 ), and the space gr r D U L(e 0 , e 1 ) = gr r D T (e 0 , e 1 ) of D-degree r is the spanned by words in e 0 , e 1 with exactly r occurrences of e 1 . Antisymmetrizing, and using Proposition 2.2, we obtain a formula for the Ihara bracket The linearized double shuffle relations on words translate into functional equations for polynomials after applying the map ρ. We describe some of these below.

Translation invariance
The additive group G a acts on A r+1 by translation, and hence G a (Q) acts on its ring of functions via λ : (y 0 , . . . , y r ) → (y 0 + λ, . . . , y r + λ). Taking the coefficients of (6.7) gives equation I2 of [Bro12], which gives a formula for the shuffleregularization of iterated integrals which begin with any sequence of 0's.
To avoid confusion, we reserve the variables x 1 , . . . , x r for the reduced polynomialρ(ξ) and use the variables y 0 , . . . , y r as above to denote the full polynomial ρ(ξ).

Antipodal symmetries
The set of primitive elements in a Hopf algebra is stable under the antipode. For the shuffle Hopf algebra this is the map * : T (e 0 , e 1 ) → T (e 0 , e 1 ) which sends w → (−1) |w|w , wherew is the reversed word and |w| the length of w. Restricting to D-degree r and transporting via ρ, we obtain a map Since the stuffle algebra, graded for the depth filtration, is isomorphic to the shuffle algebra on Y (5.9), it follows that its antipode is the map y i 1 . . . y ir → (−1) r y ir . . . and therefore if f ∈ Q[y 0 , . . . , y r ] satisfies both translational invariance and the linearized stuffle relations, we have The composition τ σ is the signed cyclic rotation of order r + 1, τ σ(f )(y 0 , . . . , y r ) = (−1) deg f f (y r , y 0 , . . . , y r−1 ), and plays an important role in the rest of this paper.

Parity relations
The following result is well known, and was first proved by Tsumura [Tsu04], and subsequently in ([IKZ06, Theorem 7]). We repeat the proof for convenience.
Proposition 6.4. The components of ls in weight N and depth r vanish unless N ≡ r mod2.
The first two terms can be interpreted as the terms occurring in the linearized stuffle product (y 2 x y 1 y 3 . . . y r ) minus the first term. As a result, one obtains the equation (y 0 , y 2 , y 1 , y 3 , . . . , y r ) + f (y 2 , y 1 , y 3 , . . . , y r , y 0 ) = 0, 551 F. Brown which, by a final application of τ σ to the right-hand term, yields y 2 , y 1 , y 3 , . . . , y r ) = 0. Therefore, in the case when deg f is odd, the polynomial f must vanish.

Dihedral symmetry and the Ihara bracket
For all r ≥ 1, consider the graded vector space p r of polynomials f ∈ Q[y 0 , . . . , y r ] which satisfy f (y 0 , . . . , y r ) = f (−y 0 , . . . , −y r ), (6.13) The maps σ, τ generate a dihedral group D r+1 = σ, τ of symmetries acting on p r of order 2r + 2, and any element f satisfying (6.13) is invariant under cyclic rotation: and anti-invariant under the reflections σ and τ . Thus Qf ⊂ p r is isomorphic to the onedimensional sign (orientation) representation ε of the dihedral group D r+1 .
Proposition 6.5. Suppose that f ∈ p r and g ∈ p s are polynomials satisfying (6.13). Then the Ihara bracket is the signed average over the dihedral symmetry group: y 1 , . . . , y r )g(y r , y r+1 . . . , y r+s ) . (6.14) In particular, {. , .} : p r × p s → p r+s , and p = r≥1 p r is a bigraded Lie algebra.
Proof. A straightforward calculation from (6.3) and definition (6.4) gives where the summation indices are taken modulo r + s + 1. Invoking dihedral symmetry of f, g leads to formula (6.14). The Jacobi identity for {. , .} is automatic since the Ihara action is an action, but can also be proved very easily by identifying its terms with the set of double cuts in a polygon (see [Bro17b]). The fact that parity (first equation of (6.13)) is preserved by { , } is clear. The anti-invariance under σ, τ is also clear from the dihedral symmetry of (6.14).
A similar dihedral symmetry was also found by Goncharov [Gon01b]; the interpretation of the dihedral reflections as antipodes may or may not be new. Since the Ihara action is, by definition, compatible with the shuffle product, it follows from Lemma 6.2 that translation invariance is preserved by the Ihara bracket. One can also easily verify this by direct computation: Definition 6.6. Letp r ⊂ p r denote the subspace of polynomials which satisfy (6.13) and are invariant under translation, and writep = r≥1p r .
Definition 6.7. We use the notation D r ⊂ Q[x 1 , . . . , x r ] to denote the spaceρ(ls r ) in depth r. It is the space of polynomial solutions to the linearized double shuffle equations in depth r and is the direct sum for all n, of spaces denoted D n,r in [IKZ06].

Generators in depth 1 and examples
It follows from Theorem 1.1 that in depth 1, the Lie algebra dg m has exactly one generator in every odd weight ≥ 3:ρ (dg m 1 ) = n≥1 Q x 2n 1 .
In particular, the algebras dg m ⊂ gr D dm 0 (Q) ⊂ ls are all isomorphic in depth 1.
Definition 6.8. Denote the Lie subalgebra generated by x 2n 1 , for n ≥ 1, by dg odd ⊂ dg m . (6.15) Example 6.9. The formula for the Ihara bracket in depth 2 can be written

Relations between depth-graded motivic multiple zeta values
We briefly explain how we may describe all relations between ζ m D using this formalism. Proposition 4.2 states that gr D H is the free bigraded right Udg m -module generated by the ζ m D (2n), and can be represented by polynomials as follows. The role of the even zeta value ζ m (2n), for n ≥ 1, is played by the depth-graded associator element τ (1) in weight 2n constructed in [Bro17a, § § 7.3 and 7.4], and is encoded by the monomial in one variable: Therefore, a relation where ξ is of depth d and weight N of the form in which the g i are the polynomial representatives of generators of dg m and k ≥ 1.

Reminders on period polynomials
We recall some definitions from [KZ84, § 1.1]. Let S 2k (SL 2 (Z)) denote the space of cusp forms of weight 2k for the full modular group SL 2 (Z).
Definition 7.1. Let n ≥ 1 and let W e 2n ⊂ Q[X, Y ] denote the vector space of homogeneous polynomials P (X, Y ) of degree 2n − 2 satisfying P (X, Y ) + P (Y, X) = 0, P(±X, ±Y ) = P (X, Y ), (7.1) The space W e 2n contains the polynomial p 2n = X 2n−2 − Y 2n−2 , and is a direct sum where S 2n is the subspace of polynomials which vanish at (X, Y ) = (1, 0). We write S = n S 2n . The Eichler-Shimura-Manin theorem gives a map which associates, in particular, an even-period polynomial to every cusp form: Explicitly, if f is a cusp form of weight 2k, the map is given by where + denotes the projection onto invariants under the involution (X, Y ) → (X, −Y ), that is, a + (X, Y ) = 1 2 (a(X, Y ) + a(X, −Y )). The three-term equation (7.2) follows from integrating around the geodesic triangle with vertices 0, 1, i∞ and is reminiscent of the hexagon equation for associators. The map (7.3) is an isomorphism onto a subspace of W e 2k ⊗ C of codimension 1. Thus dim S 2k (SL 2 (Z)) = dim S 2k and it follows from classical results on the space of modular forms of level 1 that n≥0 dim S 2n s 2n = s 12 (1 − s 4 )(1 − s 6 ) = S(s). (7.4)

F. Brown
The space D 3 consists of homogeneous polynomials f ∈ Q[x 1 , x 2 , x 3 ] such that where f (x, y, z) = f (x, x + y, x + y + z). The general double shuffle equations and their linearized versions are derived in [Bro17b, § § 4-7] using Hopf-algebra-theoretic techniques.

A short exact sequence in depth 2
The Ihara bracket gives a map { , } : ls 1 ∧ ls 1 −→ ls 2 . (7.5) Applying the isomorphismρ leads to a map given by the formula in Example 6.9. Since D 1 is isomorphic to the graded vector space Q[x 2n 1 , n ≥ 1], it follows that D 1 ∧ D 1 is isomorphic to the space of antisymmetric even polynomials p(x 1 , x 2 ) with positive bidegrees, with basis x 2m 1 x 2n 2 − x 2n 1 x 2m 2 for m > n > 0. It follows from Example 6.9 that the image of p(x 1 , x 2 ) under (7.6) is Comparing with (7.2) and (7.1), we immediately deduce (cf. [IT93,Gon05,GKZ06,Sch06]) that (7.7) In fact, the dimensions of the space D 2 have been computed several times in the literature (e.g. by some simple representation-theoretic arguments), and it is relatively easy to show [GKZ06] that the following sequence is exact: Example 7.2. The smallest non-trivial period polynomial occurs in degree 10 and is given by By the exact sequence (7.8) it immediately gives rise to the equation which, by the faithfulness of the mapρ, is equivalent to Ihara's formula (1.7).

A short exact sequence in depth 3
If The triple Ihara bracket gives a trilinear map Lie 3 (ls 1 ) −→ ls 3 , and hence a map Lie 3 (D 1 ) → D 3 whose image is spanned by {x 2a 1 , {x 2b 1 , x 2c 1 }}, for a, b, c ≥ 1. Goncharov has studied the space D 3 , and computed its dimensions in each weight [Gon05]. It follows from his work that the sequence is exact, where the first map (identifying S with ker(Λ 2 D 1 → D 2 )) is given by and the second map in the sequence immediately above is Starting from depth 4, the structure of ls d ∼ = D d is not known. 6 In particular, it is easy to show that the map given by the quadruple Ihara bracket is not surjective, since in weight 12, dim D 4 = 1, but Lie 4 (D 1 ) = 0. Our next purpose is to construct the missing elements in depth 4.
Remark 7.3. A different way to think about the sequence (7.10) is via the curious equality dim S ⊗ Q D 1 = dim Λ 3 (D 1 ) which follows from (7.4). I do not know if there is an appropriate combinatorial or modular interpretation of this identity which could be relevant to the previous exact sequence.

Linearized equations in depth 4
For the convenience of the reader, we write out the linearized double shuffle relations in full in depth 4. There are four equations. In order to write them down we shall use the following notation, where f ∈ Q[x 1 , . . . , x 4 ], and we are given any set of indices {i, j, k, l} = {1, 2, 3, 4}:
We construct some exceptional solutions to these equations from period polynomials.
Remark 8.2. The full expression forē f is explicitlȳ Since f is even it vanishes to order 2 along x = 0, y = 0, x = y. Therefore Proof. The injectivity follows immediately from (8.7). The proof that the linearized double shuffle relations hold is a finite computation. In the absence of a purely conceptual proof, we shall break the calculation into more easily verifiable pieces. We first consider the stuffle equations. It follows from general properties of the Dynkin operator on shuffle algebras that a homogeneous polynomial in four variables satisfies the two linearized stuffle equations where α(f )(x 1 , . . . , x 4 ) is the linear combination For a detailed discussion and proofs, we refer to [Bro17b,§ 16.4 and Corollary 16.6]. The linearized stuffle relations (8.2) hold for the sum of the first five terms in f 1 , and for the sum of the second five terms in f 0 in (8.6) separately. Consider first the terms in f 1 . They consist of two parts: and T 2 = f 1 (−x 4 , x 3 − x 2 ) + f 1 (x 1 , x 4 − x 3 ) + f 1 (x 2 − x 1 , −x 4 ) + f 1 (x 3 − x 2 , x 1 ).
One easily checks that λ(T 1 ) = 4 T 1 , and that λ(f 1 (x 1 , x 2 − x 3 )) equals using only the fact that f 1 is antisymmetric and odd in x and y. Thus T 1 , T 2 lie in the image of λ and are solutions to the linearized stuffle equations. Now consider the terms in f 0 in (8.6). Once again, they break into two parts: and One checks that λ(T k ) = 4 T k for k = 3, 4 using the three-term relation (8.4), and hence T 3 , T 4 are solutions to the linearized stuffle equations. This is the only point in the proof where the three-term relation is needed.
For the linearized shuffle relations, note that there exists g ∈ Q[x, y] such that f 0 (x, y) = (x + y)g(x, y) and f 1 = (x 2 − y 2 )g(x, y) since an even-period polynomial f (x, y) vanishes along x = y and is even in both x and y. The polynomial g(x, y) is symmetric and odd in x and in y (i.e. g(x, y) = g(y, x) and g(−x, y) = g(x, −y) = −g(x, y)). These properties suffice to prove that 559