Ecalle's Senary Relation in Mould Theory

• Ecalle's Senary Relation is a symmetry principle in mould theory that enforces higher-order compatibility among combinatorial transformations underlying multiple zeta values.
• It provides a bridge linking the double shuffle Lie algebra to the Kashiwara–Vergne Lie algebra through operator identities involving teru, push, and mantar.
• Its framework leverages dimorphic structures and transformations via adari(pal) and ganit(pic) to elucidate intricate algebraic relations in analytic and combinatorial number theory.

Ecalle's Senary Relation is a fundamental symmetry principle arising within Ecalle's mould-theoretic framework for multiple zeta values (MZVs) and the associated theory of double shuffle relations. This relation encapsulates a precise higher-order compatibility between the various algebro-combinatorial symmetries that govern the algebraic structures underlying MZVs. It plays a pivotal role in the construction and classification of the double shuffle Lie algebra, its injection into the Kashiwara–Vergne (KV) Lie algebra, and the general theory of "dimorphic" (dual-symmetry) structures. The Senary Relation sits at the heart of recent advances in the analytic-combinatorial approach to MZVs and their associated operadic and Lie-theoretic structures.

1. Mould Theory, Symmetries, and the Genesis of the Senary Relation

Écalle’s mould theory provides a comprehensive mechanism to encode power series, word series, and combinatorial identities through "moulds" (maps from words to a commutative ring or field) and "bimoulds" (maps from pairs of words, or 'biwords'). Central to this theory are specific symmetry constraints:

• Alternality: A mould vanishes upon any nontrivial shuffle of its arguments, reflecting the shuffle relations satisfied by iterated integral representations of MZVs.
• Symmetrality: For bimoulds, concatenation of arguments induces a multiplicative structure, emulating stuffle relations.
• Alternility/Symmetril: These encode similar conditions under conjugated or transformed arguments (e.g., via the "swap" operator).

The principal operators—swap, anti, neg, push, mantar, among others—implement involutions or cyclic actions, allowing the translation between representations and the identification of hidden symmetries.

The Senary Relation, so termed because it typically involves six components or transformations, emerges as a compatibility constraint amid these symmetry types. It appears as a "hidden extra symmetry" overlaying both shuffle and stuffle symmetries and is essential to the algebraic characterization of double shuffle relations in mould-theory language (Schneps, 2015, Kawamura, 25 Sep 2025).

2. Formal Definition and Operator Identities

While no single universal formula captures all contexts of Ecalle's Senary Relation, its canonical occurrence is as an operator identity among key transformations on moulds:

$$\operatorname{teru}(M)^\bullet = \operatorname{push} \circ \operatorname{mantar} \circ \operatorname{teru} \circ \operatorname{mantar}(M)^\bullet$$

Here, for a mould $M$, the notation $M^\bullet$ denotes the components in each depth, and the operators are as follows:

• teru: Given by

$$\operatorname{teru}(M)(u_1, ..., u_m) = M(u_1, ..., u_m) + \{ M(u_1, ..., u_{m-2}, u_{m-1}+u_m) - M(u_1, ..., u_{m-1}) \} \cdot u_m$$

• push: The cyclic permutation

$$\operatorname{push}(M)(u_1, ..., u_m) = M(u_m, u_1, ..., u_{m-1})$$

• mantar: An involutive operator, often reversing the tuple and introducing sign changes.

This identity imposes an anti-palindromicity that is characterized in Lie algebraic terms; for any depth, applying this six-step process yields, up to a possible sign, the original mould component. For various classes of moulds and associated Lie polynomials, this is equivalent to anti-palindromicity of certain derived polynomial components (Furusho et al., 2022, Kawamura, 25 Sep 2025).

A related formulation in the context of bimoulds and dimorphic structures is

$$B = (\operatorname{push} \circ \operatorname{mantar} \circ \operatorname{mantar})(B)$$

which equates a bimould to its image under a double involution and cyclic shift, expressing the core compatibility between standard and twisted symmetries (Kawamura, 25 Sep 2025).

3. Algebraic Structures and Dimorphic Symmetry

Ecalle’s Senary Relation underpins the theory of dimorphic structures—objects simultaneously satisfying a standard symmetry (such as alternality) and a "twisted" (conjugate flexion) symmetry. In the context of multiple zeta values, this means that the generating moulds for MZVs must obey both the shuffle and stuffle symmetries, as well as the senary compatibility connecting their respective dual representations.

A central aspect is the presence of "corrected" or "twisted" symmetry, realized via conjugation with certain flexion units (denoted 𝔽 or 𝔒). This duality is key to understanding the linearized picture (as in the double shuffle Lie algebra) and the nonlinear intrinsic structures (such as those realized in KV theory).

Dilators such as the gari–dilator, gira–dilator, and mu–dilator (where defined) serve to pass between exponentiated, logarithmic, and 'deformed' versions of moulds, encapsulating the structural presence of the Senary Relation within the associated Lie-theoretic operations (Kawamura, 25 Sep 2025).

4. Relation to Double Shuffle and Kashiwara–Vergne Lie Algebras

The Senary Relation governs the Lie-theoretic context in which double shuffle and Kashiwara–Vergne structures interact.

• Double Shuffle Lie Algebra ($\mathfrak{dmr}$): Generated by elements that satisfy both the shuffle (alternal) and stuffle (alternil) relations, as encoded in suitable spaces of moulds.
• Kashiwara–Vergne Lie Algebra ($\mathfrak{krv}$): Arises from special derivations on free Lie algebras in two variables, subject to anti-palindromicity and trace constraints.

Ecalle’s Senary Relation ensures the compatibility required to embed the double shuffle Lie algebra into the KV algebra. Specifically, for an element $f \in \mathfrak{dmr}$, the associated derivation $d_f$ belongs to the subalgebra of special derivations $\mathfrak{sder}$ if and only if the corresponding mould $M = \mathrm{ma}(f)$ satisfies the Senary Relation (Furusho et al., 2022, Kawamura, 25 Sep 2025). This provides a precise mould-theoretic bridge between these key algebraic structures and underpins the explicit construction of injection morphisms.

5. Intertwining Operators: adari(pal), ganit(pic), and Structural Isomorphisms

Within Ecalle’s framework, two families of operators, adari and ganit, serve to translate between the symmetry types:

• adari(pal): The adjoint action by the symmetral mould pal, transforming alternal moulds to alternil ones.
• ganit(pic): An automorphism induced via the symmetril mould pic, providing explicit isomorphisms between spaces of symmetral and symmetril moulds.

These operators commute with key group–Lie algebra maps (such as the exponential) and are algebra isomorphisms between the relevant subspaces (Komiyama, 2021). The following diagram encapsulates the higher symmetry (Senary) structure:

$$\begin{array}{ccc} \mathrm{GARI(I)_{as}} & \xrightarrow{\exp_x} & \mathrm{ARI(T)_{al}} \ \downarrow\,\mathrm{ganit(pic)} & & \downarrow\,\mathrm{ganit(pic)} \ \mathrm{GARI(I)_{is}} & \xrightarrow{\exp_x} & \mathrm{ARI(T)_{il}} \end{array}$$

Thus, the Senary Relation is realized as a canonical isomorphism, not only between different symmetry-typed spaces, but also between linear and nonlinear (group–Lie algebra) pictures.

6. Proofs, Explicit Identities, and Generalizations

Substantial effort has been devoted to formulating and proving the Senary Relation in full generality. In small depths (r = 1, 2, 3), the relation has been explicitly verified for all moulds arising from $\mathfrak{dmr}$ via direct computation involving the operators teru, push, mantar, and collision/translation maps (Furusho et al., 2022). The correctness in all depths is established in (Kawamura, 25 Sep 2025), relying on the structure of dimorphic symmetries and the full machinery of Ecalle's flexion units.

Operator-based fundamental identities, such as

$$\text{swap}\circ \text{adari(pal)}(M) = \text{ganit}_{pic}\circ \text{adari(pil)}\circ \text{swap}(M)$$

and inversion formulas involving ganit(pic) and associated push/mantar actions, provide the technical backbone enabling the Senary Relation to bridge combinatorial conditions with algebraic identities (Schneps, 2015, Komiyama, 2021).

Defining tables of operators and their mappings:

Operator	Definition/Action	Symmetry effected
adari(pal)	Adjoint by pal: $A \mapsto \mathrm{pal} \, A \, \mathrm{pal}^{-1}$	alternal → alternil
ganit(pic)	Automorphism via pic: reshuffles entries as prescribed	symmetral → symmetril
push, mantar, swap	Cyclic, reversal, variable-swap	Higher symmetries

These structures ensure closure under ari-bracket and compatibility with group operations encoding double shuffle relations.

7. Impact, Applications, and Current Directions

The Senary Relation as established enables the systematic construction of dimorphic algebraic objects and injective morphisms between double shuffle and KV Lie algebras (Kawamura, 25 Sep 2025). It clarifies why double shuffle relations suffice (in conjectural scenarios) to generate all algebraic relations among MZVs (Schneps, 2015).

Practical implications cover:

• Isomorphism theorems and explicit injections between diverse Lie algebras governing period relations.
• Understanding of resummation in analytic dynamics and averaging processes, via Rota–Baxter algebraic factorizations (as in Ecalle’s averages) (Vieillard-Baron, 2019).
• Structural frameworks for renormalization, combinatorial Hopf algebras, and the theory of arborified structures (Ebrahimi-Fard et al., 2016).

Ongoing research seeks further simplification of the foundational proofs, extension to colored or twisted versions of MZVs, and deeper exploration of transcendence and algebraic independence phenomena mediated by mould-theoretic symmetries.

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Ecalle's Senary Relation represents a unifying symmetry principle in the mould-theoretic language governing the algebraic structure of multiple zeta values, bridging the classical and modern perspectives on the interplay between double shuffle relations, Lie theory, and the combinatorics underlying fundamental periods in arithmetic geometry and quantum algebra.