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Symmetry on Linear Relations for Multiple Zeta Values

Published online by Cambridge University Press:  11 January 2016

Kentaro Ihara  and
Hiroyuki Ochiai
Kentaro Ihara
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama, Toyonaka Osaka 560-0043, Japan, [email protected]
Hiroyuki Ochiai
Affiliation:
Department of Mathematics, Nagoya University, Chikusa, Nagoya 464-8602, Japan, [email protected]
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Abstract

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We find a symmetry for the reflection groups in the double shuffle space of depth three. The space was introduced by Ihara, Kaneko and Zagier and consists of polynomials in three variables satisfying certain identities which are connected with the double shuffle relations for multiple zeta values. Goncharov has defined a space essentially equivalent to the double shuffle space and has calculated the dimension. In this paper we relate the structure among multiple zeta values of depth three with the invariant theory for the reflection groups and discuss the dimension of the double shuffle space in this view point.

Keywords

11M4111M0640B05
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 189 , 2008 , pp. 49 - 62
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2008

References

[1] Broadhurst, D. J. and Kreimer, D., Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Physics Lett. B, 393 (1997), 403–412.Google Scholar
[2] Goncharov, A. B., Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett., 5 (1998), 497–516.Google Scholar
[3] Goncharov, A. B., Multiple (-values, Galois groups and geometry of modular varieties, Progr. Math. 201, Birkhäuser, 2001, pp. 361–392.Google Scholar
[4] Goncharov, A. B., The dihedral Lie algebras and Galois symmetries of π1 (l) (ℙ1- ({0,∞}∪ μN)) , Duke Math. J., 110 (2001), 397–487.Google Scholar
[5] Goncharov, A. B., Multiple polylogarithms and mixed Tate motives, preprint (2001), math.AG/0103059.Google Scholar
[6] Humphreys, J. E., Reflection groups and Coxeter Groups, Cambridge, 1990.Google Scholar
[7] Ihara, K., Derivations and double shuffle relations for multiple zeta values, RIMS Kokyuroku, 1549 (2007), 47–63.Google Scholar
[8] Ihara, K., Kaneko, M., Zagier, D., Derivations and double shuffle relations for multiple zeta values, Compositio Math., 142 (2006), 307–338.CrossRefGoogle Scholar
[9] Zagier, D., Periods of modular forms, trace of Hecke operators, and multiple zeta values, RIMS Kokyuroku, 843 (1993), 162–170.Google Scholar
[10] Zagier, D., Values of zeta functions and their applications, Progr. Math. 120, Birkhäuser, 1994, pp. 497–512.Google Scholar