Evaluation of the multiple zeta values $\zeta(2,\ldots,2,4,2,\ldots,2)$ and period polynomial relations

TL;DR

This paper investigates specific multiple zeta values and their relations to period polynomials, revealing new symmetries and evaluations that connect multiple zeta values with classical double zeta values.

Contribution

It provides explicit evaluations of certain multiple zeta values as polynomials in double zeta values and uncovers intrinsic symmetries linking block graded multiple zeta values to period polynomial relations.

Findings

Derived period polynomial relations from explicit evaluations.

Established Galois descent of alternating double zeta values.

Expressed multiple t values in terms of classical double zeta values.

Abstract

In studying the depth filtration on multiple zeta values, difficulties quickly arise due to a disparity between it and the coradical filtration. In particular, there are additional relations in the depth graded algebra coming from period polynomials of cusp forms for $SL_2(\mathbb{Z})$. In contrast, a simple combinatorial filtration, the block filtration is known to agree with the coradical filtration, and so there is no similar defect in the associated graded. However, via an explicit evaluation of $\zeta(2,\ldots,2,4,2,\ldots,2)$ as a polynomial in double zeta values, we derive these period polynomial relations as a consequence of an intrinsic symmetry of block graded multiple zeta values in block degree 2. In deriving this evaluation, we find a Galois descent of certain alternating double zeta values to classical double zeta values, which we then apply to give an evaluation of the multiple $t$ values $t(2\ell,2k)$ in terms of classical double zeta values.