Analogues of Cyclic Insertion-Type Identities for Multiple Zeta Star Values

Charlton, Steven

doi:10.2206/kyushujm.74.337

Cited by 2 publications

(4 citation statements)

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“…In particular, we need to use the stuffle antipode ([24], [31] or [18, 19]) to convert to a corresponding multiple zeta star value. Then we can apply Zhao’s generalised 2-1 Theorem [41] (in the block decomposition form [12] for convenience) to rewrite the zeta star value as an alternating zeta-half value. By application of the parity theorem ([33], or [21]), we reduce this to an explicit combination of depth 2 alternating MZVs.…”

Section: Alternating and Interpolated Mzvsmentioning

confidence: 99%

“…Then where if and if , and if , one should neglect this argument. This follows by combining Zhao’s generalised 2-1 Theorem [41], which involves a certain recursively constructed sequence of indices , with the description of the final such index string , given in [12, Lemma 3.1], in terms of the block decomposition. This final string supplies the arguments in Zhao’s formulation of the 2-1 Theorem. In our case, we want to apply this to .…”

Section: Alternating and Interpolated Mzvsmentioning

confidence: 99%

“…We now recall the generalised 2-1 Theorem, established by Zhao [41], which evaluates each value in terms of a certain alternating value. It is more convenient – and indeed has a closer connection with the goal – to write the generalised 2-1 Theorem in the block decomposition form given in [12, Lemma 3.1].…”

Section: Alternating and Interpolated Mzvsmentioning

confidence: 99%

“…Specifically, an evaluation of in terms of double zeta values 1 , and a computation of the dimension of the space cut out by the block dihedral symmetry. The latter is a straightforward argument from representation theory, while the former is a trek through the world of MZV relations and machinery, including: motivic cobracket calculations; multiple Euler sums (also called alternating MZVs) and the octagon-relation-induced dihedral symmetries thereof [18, 19]; multiple zeta star values and the stuffle-antipode [31, Equation 2.4],[24, Proposition 1]; Zhao’s generalised 2-1 theorem [41] (and the first author’s block-decomposition description thereof [12]); (alternating) multiple zeta-half values [39]; the explicit depth-parity theory for depth 3 alternating MZVs [21, 33]; and a vital (and serendipitously unearthed) generalised doubling relation [42, Section 14.2.5]. We divide these results between Section 4 and an appendix, according to how central they are to the main results of the article.…”

Section: Introductionmentioning

confidence: 99%

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Evaluation of and period polynomial relations

Charlton,

Keilthy

2024

Forum of Mathematics, Sigma

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In studying the depth filtration on multiple zeta values, difficulties quickly arise due to a disparity between it and the coradical filtration [9]. In particular, there are additional relations in the depth graded algebra coming from period polynomials of cusp forms for $\operatorname {\mathrm {SL}}_2({\mathbb {Z}})$ . In contrast, a simple combinatorial filtration, the block filtration [13, 28] 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 [22] $t(2\ell ,2k)$ in terms of classical double zeta values.

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