Standard relations of multiple polylogarithm values at roots of unity
Let \[ \text{Li}_{s_1,\dots,s_\ell}(x_1,\dots,x_\ell) =\sum_{n_1>\dots>n_\ell\geq1}\frac{x_1^{n_1}\dotsb x_\ell^{n_\ell}}{n_1^{s_1}\dotsb n_\ell^{s_\ell}} \] denote the generalized multiple polylogarithm. The quantities \(\ell\) and \(w=s_1+\dots+s_\ell\) are called the \textit{depth} (or \textit{length}) and the \textit{weight}, respectively. For a given integer \(N>0\), fix an \(N\)th root of unity \(\mu=\exp(2\pi\sqrt{-1}/N)\) and consider the \textit{multiple polylogarithm value} (MPV) \[ L_N(s_1,\dots,s_\ell\mid i_1,\dots,i_\ell) =\text{Li}_{s_1,\dots,s_\ell}(\mu^{i_1},\dots,\mu^{i_\ell}) \] \textit{of level}~\(N\); the defining series converges if and only \((s_1,\mu^{i_1})\neq(1,1)\). The MPVs of the same level and weight are known to satisfy the \textit{standard} algebraic relations which include the regularized double shuffle relations, regularized distribution relations, lifted versions of such relations from lower weights, and weight one relations which are produced by relations of weight one MPVs. Denote by \(d(w,N)\) the dimension of the \(\mathbb Q\)-vector space spanned by all MPVs of weight \(w\) and level~\(N\). Recently \textit{P.~Deligne} and \textit{A.~Goncharov} [Ann. Sci. Éc. Norm. Supér. (4) 38, No.~1, 1--56 (2005; Zbl 1084.14024)] were able to obtain some lower bound of \(d(w,N)\) using the motivic mechanism. The computation done in the paper under review suggests that, for \(N\) of the form \(N=1,2,3\) or \(N=p^n\) for prime \(p\geq5\), the standard relations should produce all the linear relations and, if further \(N=p^n\) for \(p\geq5\), then the bound of \(d(w,N)\) by Deligne and Goncharov can be improved; otherwise there should be non-standard relations among MPVs for all sufficiently large weights (depending only on~\(N\)) and the bound by Deligne and Goncharov may be sharp. The author indicates explicitly some of non-standard relations, with good numerical verification, and in two cases, \(N=4\) and \(w=3,4\), proves these relations rigorously by using the octahedral symmetry of \(\{0,\infty,\pm1,\pm\sqrt{-1}\}\). In addition, the author provides many conjectures which are strongly supported by computational evidence.
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