On the Kernel of the Symbol Map for Multiple Polylogarithms

Abstract

The symbol map (of Goncharov) takes multiple polylogarithms to a tensor product space where calculations are easier, but where important differential and combinatorial properties of the multiple polylogarithm are retained. Finding linear combinations of multiple polylogarithms in the kernel of the symbol map is an effective way to attempt finding functional equations. We present and utilise methods for finding new linear combinations of multiple polylogarithms (and specifically harmonic polylogarithms) that lie in the kernel of the symbol map.

During this process we introduce a new pictorial construction for calculating the symbol, namely the hook-arrow tree, which can be used to easier encode symbol calculations onto a computer.

We also show how the hook-arrow tree can simplify symbol calculations where the depth of a multiple polylogarithm is lower than its weight and give explicit expressions for the symbol of depth 2 and 3 multiple polylogarithms of any weight. Using this we give the full symbol for I_(x,y,z). Through similar methods we also give the full symbol of coloured multiple zeta values.

We provide introductory material including the binary tree (of Goncharov) and the polygon dissection (of Gangl, Goncharov and Levin) methods of finding the symbol of a multiple polylogarithm, and give bijections between (adapted forms of) these methods and the hook-arrow tree.