Schemes of double shuffle and distribution relations among cyclotomic multiple zeta values

Abstract: We are interested in two formal approaches reflecting the combinatorial properties of double shuffle relations between cyclotomic multiple zeta values of level \(N \ge 1\). The first approach, introduced by Racinet, considers cyclotomic multiple zeta values from the perspective of the Drinfeld associator and provides a description based on Hopf algebra coproducts, which he encodes in a scheme DMR(N). The second, studied by Hoffmann, Ihara-Kaneko-Zagier (\(N=1\)), Arakawa-Kaneko and Zhao (\(N \ge 1\)), describes these relations through algebra products that we encode in a scheme EDS(N). When \(N > 1\), the cyclotomic multiple zeta values of level N also satisfy distribution relations that Racinet incorporates into a subscheme DMRD(N) of DMR(N). In this presentation, we establish an isomorphism between the schemes DMR(N) and EDS(N), then introduce a subscheme EDSD(N) of EDS(N) that we identify with DMRD(N). This identification enables us to prove a conjecture of Zhao stating that the weight 2 distribution relations are a consequence of double shuffle relations as well as weight 1 and depth 2 distribution relations (this talk is based on a joint work with Henrik Bachmann).

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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