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A character on the quasi-symmetric functions coming from multiple zeta values. (English) Zbl 1163.05334
Summary: We define a homomorphism \(\zeta\) from the algebra of quasi-symmetric functions to the reals which involves the Euler constant and multiple zeta values. Besides advancing the study of multiple zeta values, the homomorphism \(\zeta\) appears in connection with two Hirzebruch genera of almost complex manifolds: the \(\Gamma\)-genus (related to mirror symmetry) and the \(\hat{\Gamma}\)-genus (related to an \(S^1\)-equivariant Euler class). We decompose \(\zeta\) into its even and odd factors in the sense of Aguiar, Bergeron, and Sottille, and demonstrate the usefulness of this decomposition in computing \(\zeta\) on the subalgebra of symmetric functions (which suffices for computations of the \(\Gamma\)- and \(\hat{\Gamma}\)-genera).

05E05 Symmetric functions and generalizations
11M32 Multiple Dirichlet series and zeta functions and multizeta values
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
57R20 Characteristic classes and numbers in differential topology
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