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Báo cáo toán học: "A character on the quasi-symmetric functions coming from multiple zeta values"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A character on the quasi-symmetric functions coming from multiple zeta values. | A character on the quasi-symmetric functions coming from multiple zeta values Michael E. Hoffman Dept. of Mathematics U. S. Naval Academy Annapolis MD 21402 USA meh@usna.edu Submitted May 6 2008 Accepted Jul 23 2008 Published Jul 28 2008 Keywords multiple zeta values symmetric functions quasi-symmetric functions Hopf algebra character gamma function r-genus r-genus Mathematics Subject Classifications Primary 05E05 Secondary 11M41 14J32 16W30 57R20 Abstract We define a homomorphism 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 appears in connection with two Hirzebruch genera of almost complex manifolds the T-genus related to mirror symmetry and the T-genus related to an S1-equivariant Euler class . We decompose into its even and odd factors in the sense of Aguiar Bergeron and Sottille and demonstrate the usefulness of this decomposition in computing on the subalgebra of symmetric functions which suffices for computations of the T-and T-genera . 1 Introduction Let x1 x2 . be a countably infinite sequence of indeterminates each having degree 1 and let P c R x1 x2 . be the set of formal power series in the Xj having bounded degree. Then P is a graded algebra over the reals. An element f 2 P is called a symmetric function if coefficient of xýXn2 xý in f coefficient of xl1 x22 xk in f 1 for any k-tuple n1 . nk of distinct positive integers and f is called a quasi-symmetric function if equation 1 holds whenever H1 n2 nk. The vector spaces Sym and QSym of symmetric and quasi-symmetric functions respectively are both subalgebras of P with Sym c QSym. Of course Sym is a familiar object THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R97 1 for which the first chapter of Macdonald 20 is a convenient reference. The algebra QSym was introduced by Gessel 9 and in recent years has become increasingly important in combinatorics see e.g. 24 .