A Character on the Quasi-Symmetric Functions coming from Multiple Zeta Values

Michael E. Hoffman


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).

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