Period polynomial relations between double zeta values

TL;DR

This paper explores the connections between period polynomial relations in the double shuffle Lie algebra and linear relations among double zeta values, revealing how these two sets of relations are interconnected through duality.

Contribution

It establishes a link between even weight period polynomial relations and odd-component double zeta value relations, showing they can be derived from each other via duality.

Findings

Relation between period polynomial relations and double zeta value relations established

Duality is used to deduce one set of relations from the other

Provides a unified understanding of these algebraic and number-theoretic relations

Abstract

The even weight period polynomial relations in the double shuffle Lie algebra $\mathfrak{ds}$ were discovered by Ihara, and completely classified by the second author by relating them to restricted even period polynomials associated to cusp forms on $\mathrm{SL}_2(\mathbb{Z})$. In an article published in the same year, Gangl, Kaneko and Zagier displayed certain linear combinations of odd-component double zeta values which are equal to scalar multiples of simple zeta values in even weight, and also related them to restricted even period polynomials. In this paper, we relate the two sets of relations, showing how they can be deduced from each other by duality.