From polygons and symbols to polylogarithmic functions

Details From polygons and symbols to polylogarithmic functions From polygons and symbols to polylogarithmic functions

2011-10-03

arxiv.org/abs/1110.0458v1

Physics

Mathematical Physics

75 pages. Mathematica files with the expression of all HPLs up to weight 4 in terms of the spanning set are included

Scientific paper

We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illustrate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of non-trivial elements in the kernel of the symbol map for arbitrary weight.

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