Double Shuffle Research Articles

Crossed Product Interpretation of the Double Shuffle Lie Algebra Attached to a Finite Abelian Group

Racinet studied a scheme associated with the double shuffle and regularization relations between multiple polylogarithm values at N th roots of unity and constructed a group scheme attached to the situation; he also showed it to be the specialization for G=\mu_{N} of a group scheme \operatorname{\mathsf{DMR}}_{0}^{G} attached to a finite abelian group G . Then Enriquez and Furusho proved that \operatorname{\mathsf{DMR}}_{0}^{G} can be essentially identified with the stabilizer of a coproduct element arising in Racinet’s theory with respect to the action of a group of automorphisms of a free Lie algebra attached to G . We reformulate Racinet’s construction in terms of crossed products. Racinet’s coproduct can then be identified with a coproduct \widehat{\Delta}^{\mathcal{M}}_{G} defined on a module \widehat{\mathcal{M}}_{G} over an algebra \widehat{\mathcal{W}}_{G} , which is equipped with its own coproduct \widehat{\Delta}^{\mathcal{W}}_{G} , and the group action on \widehat{\mathcal{M}}_{G} extends to a compatible action of \widehat{\mathcal{W}}_{G} . We then show that the stabilizer of \widehat{\Delta}^{\mathcal{M}}_{G} , hence \operatorname{\mathsf{DMR}}_{0}^{G} , is contained in the stabilizer of \widehat{\Delta}^{\mathcal{W}}_{G} thus generalizing a result of Enriquez and Furusho [Selecta Math. (N.S.) 29 (2023), article no. 3]. This yields an explicit group scheme containing \operatorname{\mathsf{DMR}}_{0}^{G} , which we also express in the Racinet formalism.

Has ‘the great moving right show’ stopped?

How will the new Labour government compare to its New Labour predecessors? This article revisits ‘New Labour’s Double Shuffle’, the critique of the New Labour government that Stuart Hall wrote in 2003, to see how this might provide guidance in understanding the conjuncture of the Labour government elected in July 2024. Hall???s article had followed on from his pre-election analysis of the Thatcher project in ‘The Great Moving Right Show’, in 1978. His argument was that the dominant agenda of the New Labour governments was one of subordination to pressures emanating from capital, with the demands of Labour’s supporters being subordinated to this. It is proposed that the relations between the dominant and subordinate elements in these structures should be seen in dialectical terms, as an ongoing struggle between conflicting social forces. The question is raised of whether the current Labour government is essentially a continuation of the New Labour project, or whether it might be something different. The argument is made that to achieve a favourable outcome in a new struggle for hegemony it is essential that Labour remain in office for several parliamentary terms, and that there be a critical engagement with what its government does.

Alternating multiple mixed values: Regularization, special values, parity, and dimension conjectures

In this paper, we define and study a variant of multiple zeta values (MZVs) of level four, called alternating multiple mixed values or alternating multiple M-values (AMMVs), forming a Q[i]-subspace of the colored MZVs of level four. This variant includes the alternating version of Hoffman’s multiple t-values, Kaneko–Tsumura’s multiple T-values, and the multiple S-values studied by the authors previously as special cases. We exhibit nice properties similar to the ordinary MZVs such as the generalized duality, integral shuffle and series stuffle relations. After setting up the algebraic framework we derive the regularized double shuffle relations of the AMMVs by adopting the machinery from color MZVs of level four. As an important application, we prove a parity result for AMMVs previously conjectured by two of us. We also investigate several alternating multiple S- and T-values by establishing some explicit relations of integrals involving arctangent function. At the end, we compute the dimensions of a few interesting subspaces of AMMVs for weight less than 9. Supported by theoretical and numerical evidence aided by numerical and symbolic computation, we formulate a few conjectures concerning the dimensions of the above-mentioned subspaces of AMMVs. These conjectures hint at a few very rich but previously overlooked algebraic and geometric structures associated with these vector spaces.

Ideology, statecraft and the ‘double shuffle’ of Conservative planning reform in England

This article explores the political implications of conflicts over new housebuilding for Conservative-led governments in England since 2010. Revisiting debates about the tensions between neoliberal and more collectivist traditions within the political ideology of the Conservative Party, we argue that rather than blocking change this internal conflict should be seen as a dynamic part of the politics of planning. This leads us to propose that Conservative planning reforms have been characterised by a distinctive ‘double shuffle’ through which the party has sought to progress neoliberalising reforms whilst managing these tensions and seeking to maintain their hold on power through statecraft. This article was published open access under a CC BY licence: https://creativecommons.org/licences/by/4.0/

The cyclotomic double shuffle torsor in terms of Betti and de Rham coproducts

We prove a cyclotomic generalization of Enriquez and Furusho's result stating that the scheme DMR× of double shuffle and regularization relations between multiple zeta values arising from Racinet's formalism is a torsor of isomorphisms between “Betti” and “de Rham” objects. After reviewing the “de Rham” objects arising from a crossed product interpretation of Racinet's formalism, we construct the “Betti” objects based on the orbifold fundamental group of the quotient (C×∖μN)/μN with μN being the group of Nth roots of unity for N≥1. We then show that DMR× is a subtorsor of stabilizer schemes relating “Betti” and “de Rham” coproducts.

Combinatorial multiple Eisenstein series

We construct a family of q-series with rational coefficients satisfying a variant of the extended double shuffle equations, which are a lift of a given mathbb {Q}-valued solution of the extended double shuffle equations. We call these q-series combinatorial (bi-)multiple Eisenstein series, and in depth one they coincide with (classical) Eisenstein series. Combinatorial multiple Eisenstein series can be seen as an interpolation between the given mathbb {Q}-valued solution of the extended double shuffle equations (as qrightarrow 0) and multiple zeta values (as qrightarrow 1). In particular, they are q-analogues of multiple zeta values closely related to modular forms. Their definition is inspired by the Fourier expansion of multiple Eisenstein series introduced by Gangl-Kaneko-Zagier. Our explicit construction is done on the level of their generating series, which we show to be a so-called symmetril and swap invariant bimould.

Realizations of the double shuffle relations

The set of multiple zeta values satisfy the double shuffle relations. In this paper, we study potential realizations of the double shuffle relations from multiple series and iterated path integrals. We show that the realizations of the double shuffle relations which can be constructed from multiple series are unique up to multiplying by a constant. We prove that there are no nontrivial realizations of the double shuffle relations from iterated path integrals of continuous functions on closed intervals. Lastly we show that the set of one variable multiple polylogarithms satisfies the double shuffle relations only for the point 0 and 1.

The Betti side of the double shuffle theory. III. Bitorsor structures

In the first two parts of the series, we constructed stabilizer subtorsors of a ‘twisted Magnus’ torsor, studied their relations with the associator and double shuffle torsors, and explained their ‘de Rham’ nature. In this paper, we make the associated bitorsor structures explicit and explain the ‘Betti’ nature of the corresponding right torsors; we thereby complete one aim of the series. We study the discrete and pro-p versions of the ‘Betti’ group of the double shuffle bitorsor.

Bigraded Lie Algebras Related to Multiple Zeta Values

We prove that the dihedral Lie coalgebra D_{\bullet \bullet}:=\bigoplus_{k\geq m \geq 1} D_{m,k} corresponding to {\widehat{\mathscr{D}}_{\bullet \bullet}(G)} of Goncharov (Duke Math. J. 110 (2001), 397–487) for G=\{e\} is the bigraded dual of the linearized double shuffle Lie algebra \mathfrak{ls}:=\bigoplus_{k\geq m \geq 1}\mathfrak{ls}_m^k\subset \mathbb{Q}\langle x,z \rangle of Brown (Compos. Math. 157 (2021), 529–572) whose Lie bracket is the Ihara bracket initially defined over \mathbb{Q}\langle x,z \rangle , by constructing an explicit isomorphism of bigraded Lie coalgebras D_{\bullet \bullet} \to \mathfrak{ls}^\vee , where \mathfrak{ls}^\vee is the Lie coalgebra dual (in the bigraded sense) to \mathfrak{ls} . The work leads to the equivalence between the two statements “ D_{\bullet \bullet} is a Lie coalgebra with respect to Goncharov’s cobracket formula in Goncharov (Duke Math. J. 110 (2001), 397–487)” and “ \mathfrak{ls} is preserved by the Ihara bracket''. We also prove folklore results from Brown (Compos. Math. 157 (2021), 529–572) and Ihara et al. (Compos. Math. 142 (2006), 307–338) (which apparently have no written proofs in the literature) stating that for m \geq 2 , D_{m,\bullet}:=\bigoplus_{k\geq m} D_{m,k} is graded isomorphic (dual) to the double shuffle space \mathrm{Dsh}_m:=\bigoplus_{k\geq m} \mathrm{Dsh}_{m}({{k}-m}) \subset \mathbb{Q}[x_1,\dots,x_m] (stated in Ihara et al., Compos. Math. \textbf{142} (2006), 307–338), and that the linear map f_m\colon \mathbb{Q}\langle x,z \rangle_m \to \mathbb{Q}[x_1,\dots,x_m] , where \mathbb{Q}\langle x,z \rangle_m is the space linearly generated by monomials of \mathbb{Q}\langle x,z \rangle of degree m with respect to z , given by x^{n_1}z\cdots x^{n_m}zx^{n_{m+1}}\mapsto \delta_{0,n_{m+1}} x_1^{n_1}\cdots x_{n_m}^{n_m} , with \delta_{a,b} the Kronecker delta, restricts to a graded isomorphism \bar{f}_m\colon \mathfrak{ls}_m:=\bigoplus_{k\geq m} \mathfrak{ls}_m^k \to \mathrm{Dsh}_{m} (stated in Brown (Compos. Math. 157 (2021), 529–572)). Here, we establish three explicit compatible isomorphisms D_{\bullet \bullet} \to \mathfrak{ls}^\vee, D_{m\bullet}\to \mathrm{Dsh}_{m}^\vee and \bar{f}_m\colon \mathfrak{ls}_m \to \mathrm{Dsh}_{m} , where \mathrm{Dsh}_{m}^\vee is the graded dual of \mathrm{Dsh}_{m} .

The Betti side of the double shuffle theory. II. Double shuffle relations for associators

We derive from the compatibility of associators with the module harmonic coproduct, obtained in Part I of the series, the inclusion of the torsor of associators into that of double shuffle relations, which completes one of the aims of this series. We define two stabilizer torsors using the module and algebra harmonic coproducts from Part I. We show that the double shuffle torsor can be described using the module stabilizer torsor, and that the latter torsor is contained in the algebra stabilizer torsor.

Algebraic relations of interpolated multiple zeta values

Interpolated multiple zeta values can be regarded as interpolation polynomials of multiple zeta values and multiple zeta-star values. In this paper, we give some algebraic relations of interpolated multiple zeta values, such as the shuffle regularized sum formula, a weighted sum formula and some evaluation formulas with even arguments. All the algebraic relations provided in this paper are deduced from the extended double shuffle relations.

Variants of multiple zeta values with even and odd summation indices

In this paper, we define and study a variant of multiple zeta values of level 2 (which is called multiple mixed values or multiple $M$-values, MMVs for short), which forms a subspace of the space of alternating multiple zeta values. This variant includes both Hoffman's multiple $t$-values and Kaneko-Tsumura's multiple $T$-values as special cases. We set up the algebra framework for the double shuffle relations (DBSFs) of the MMVs, and exhibits nice properties such as duality, integral shuffle relation, series stuffle relation, etc., similar to ordinary multiple zeta values. Moreover, we study several $T$-variants of Kaneko-Yamamoto type multiple zeta values by establishing some explicit relations between these $T$-variants and Kaneko-Tsumura $\psi$-values. Furthermore, we prove that all Kaneko-Tsumura $\psi$-values can be expressed in terms of Kaneko-Tsumura multiple $T$-values by using multiple associated integrals, and find some duality formulas for Kaneko-Tsumura $\psi$-values. We also discuss the explicit evaluations for a kind of MMVs of depth two and three by using the method of contour integral and residue theorem. Finally, we investigate the dimensions of a few interesting subspaces of MMVs for small weights.

The Betti side of the double shuffle theory. I. The harmonic coproducts

This paper is the first in a series which aims at: (a) giving a proof that the associator relations between multizeta values imply the double shuffle and regularization (DSR) ones, alternative to that of the second-named author’s 2010 paper; (b) enhancing Racinet’s construction of a torsor structure over the \({\mathbb {Q}}\)-scheme of DSR relations to an explicit bitorsor structure. In this paper, we revisit Racinet’s original DSR formalism, whose main character is an algebra coproduct, called the harmonic coproduct, and we introduce a variant which is a module coproduct; we explain the ‘de Rham’ nature of this formalism and construct a ‘Betti’ counterpart of it; we show how both formalisms can be interpreted in terms of geometry, following the ideas of Deligne and Terasoma’s unfinished 2005 preprint; we use Bar-Natan’s interpretation of associators as functors from the category of parenthesized braids to that of chord diagrams to show that any associator relates the Betti and de Rham geometric objects, both in the ‘algebraic’ and in the ‘module’ setups; we derive that any associator relates the Betti and de Rham algebra coproducts, as well as their module counterparts. These results will be used in the next parts of the series.

Medical Image Protection Algorithm Based on Deoxyribonucleic Acid Chain of Dynamic Length

Current image encryption algorithms have various deficiencies in effectively protecting medical images with large storage capacity and high pixel correlation. This article proposed a new image protection algorithm based on the deoxyribonucleic acid chain of dynamic length, which achieved image encryption by DNA dynamic coding, generation of DNA dynamic chain, and dynamic operation of row chain and column chain. First, the original image is encoded dynamically according to the binary bit from a pixel, and the DNA sequence matrix is scrambled. Second, DNA sequence matrices are dynamically segmented into DNA chains of different lengths. After that, row and column deletion operation and transposition operation of DNA dynamic chain are carried out, respectively, which made DNA chain matrix double shuffle. Finally, the encrypted image is got after recombining DNA chains of different lengths. The proposed algorithm was tested on a list of medical images. Results showed that the proposed algorithm showed excellent security performance, and it is immune to noise attack, occlusion attack, and all common cryptographic attacks.

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.

Some properties of double shuffles of bivariate copulas and (extreme) copulas invariant with respect to Lüroth double shuffles

Considering the well-known shuffling operation in x- and in y-direction yields so-called double shuffles of bivariate copulas. We study continuity properties of the double shuffle operator ST induced by pairs T=(T1×T2) of measure preserving transformations on ([0,1],B([0,1]),λ) on the family C of all bivariate copulas, analyze its interrelation with the star/Markov product, and show that for each left- and for each right-invertible copula A the set of all possible double shuffles of A is dense in C with respect to the uniform metric d∞. After deriving some general properties of the set ΩT of all ST-invariant copulas we focus on the situation where T1,T2 are strongly mixing and show that in this case the product copula Π is an extreme point of ΩT. Moreover, motivated by a recent paper by Horanská and Sarkoci (Fuzzy Sets and Systems 378, 2018) we then study double shuffles induced by pairs of so-called Lüroth maps and derive various additional properties of ΩT, including the surprising fact that ΩT contains uncountably many extreme points which (interpreted as doubly stochastic measures) are pairwise mutually singular with respect to each other and which allow for an explicit construction.

Truncated t-adic symmetric multiple zeta values and double shuffle relations

We study a refinement of the symmetric multiple zeta value, called the t-adic symmetric multiple zeta value, by considering its finite truncation. More precisely, two kinds of regularizations (harmonic and shuffle) give two kinds of the t-adic symmetric multiple zeta values, thus we introduce two kinds of truncations correspondingly. Then we show that our truncations tend to the corresponding t-adic symmetric multiple zeta values, and satisfy the harmonic and shuffle relations, respectively. This gives a new proof of the double shuffle relations for t-adic symmetric multiple zeta values, first proved by Jarossay. In order to prove the shuffle relation, we develop the theory of truncated t-adic symmetric multiple zeta values associated with 2-colored rooted trees. Finally, we discuss a refinement of Kaneko–Zagier’s conjecture and the t-adic symmetric multiple zeta values of Mordell–Tornheim type.

Some relations deduced from regularized double shuffle relations of multiple zeta values

It is conjectured that the regularized double shuffle relations give all algebraic relations among the multiple zeta values, and hence all other algebraic relations should be deduced from the regularized double shuffle relations. In this paper, we provide as many as the relations which can be derived from the regularized double shuffle relations, for example, the weighted sum formula of Guo and Xie, some evaluation formulas with even arguments and the restricted sum formulas of Hoffman and their generalizations.

Governmental social work (part 1): the tyranny of transformation

Governmental social work refers to a new ‘settlement’ for social work and social work education. A critical discourse analysis of Putting children first (Department for Education, 2016) – considered the foundational text of governmental social work – is undertaken. The analysis suggests the ways in which the transformative strategy of governmental social work seeks to achieve outcomes or objectives within existing structures and practices, especially by changing them in particular ways. Social workers are called on to become free progressive professionals as long as they comply with the form of professionalism that is legitimated by governmental social work. The reforms are represented as the only morally and professionally right responses for those who care about children. This involves a double shuffle: a process of de-professionalisation and re-professionalisation that involves identity change and subjugation for social workers in a compliant profession that increasingly ‘governs itself’ in the required ways and maintains a silence on the circumstances of children’s lives.

Elliptic double shuffle, Grothendieck–Teichmüller and mould theory

In this article we define an elliptic double shuffle Lie algebra \(\scriptstyle {{\mathfrak {ds}}_{ell}}\) that generalizes the well-known double shuffle Lie algebra \(\scriptstyle {{\mathfrak {ds}}}\) to the elliptic situation. The double shuffle, or dimorphic, relations satisfied by elements of the Lie algebra \(\scriptstyle {{\mathfrak {ds}}}\) express two families of algebraic relations between multiple zeta values that conjecturally generate all relations. In analogy with this, elements of the elliptic double shuffle Lie algebra \(\scriptstyle {{\mathfrak {ds}}_{ell}}\) are Lie polynomials having a dimorphic property called \(\scriptstyle {\Delta }\)-bialternality that conjecturally describes the (dual of the) set of algebraic relations between elliptic multiple zeta values, which arise as coefficients of a certain elliptic generating series (constructed explicitly in Lochak et al. [15]) in On elliptic multiple zeta values 2016, in preparation) and closely related to the elliptic associator defined by Enriquez [10]. We show that one of Ecalle’s major results in mould theory can be reinterpreted as yielding the existence of an injective Lie algebra morphism \(\scriptstyle {{\mathfrak {ds}}\rightarrow {\mathfrak {ds}}_{ell}}\). Our main result is the compatibility of this map with the tangential-base-point section \(\scriptstyle {\mathrm{Lie}\,\pi _1(MTM)\rightarrow \mathrm{Lie}\,\pi _1(MEM)}\) constructed by Hain and Matsumoto [14] and with the section \(\scriptstyle {{\mathfrak {grt}}\rightarrow {\mathfrak {grt}}_{ell}}\) mapping the Grothendieck–Teichmüller Lie algebra \(\scriptstyle {{\mathfrak {grt}}}\) into the elliptic Grothendieck–Teichmüller Lie algebra \(\scriptstyle {{\mathfrak {grt}}_{ell}}\) constructed by Enriquez. This compatibility is expressed by the commutativity of the following diagram (excluding the dotted arrow, which is conjectural).