Quasi-symmetric functions, multiple zeta values, and rooted trees
نویسنده
چکیده
The algebra Sym of symmetric functions is a proper subalgebra of QSym: for example, M11 and M12 +M21 are symmetric, but M12 is not. As an algebra, QSym is generated by those monomial symmetric functions corresponding to Lyndon words in the positive integers [11, 6]. The subalgebra of QSym ⊂ QSym generated by all Lyndon words other than M1 has the vector space basis consisting of all monomial symmetric functions Mp1p2···pk with pk > 1 (together with M∅ = 1). There is a homorphism QSym 0 → R given by sending each ti to 1 i ; that is, the monomial quasi-symmetric function Mp1···pk is sent to the multiple zeta value
منابع مشابه
A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees
In this paper, we construct explicitly a noncommutative symmetric (NCS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the NCS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a ...
متن کاملA Character on the Quasi-Symmetric Functions coming from Multiple Zeta Values
We define a homomorphism ζ 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 ζ appears in connection with two Hirzebruch genera of almost complex manifolds: the Γ-genus (related to mirror symmetry) and the Γ̂-genus (related to an S1-equivariant Euler class). ...
متن کاملQuasi-symmetric functions and mod p multiple harmonic sums
We present a number of results about (finite) multiple harmonic sums modulo a prime, which provide interesting parallels to known results about multiple zeta values (i.e., infinite multiple harmonic series). In particular, we prove a “duality” result for mod p harmonic sums similar to (but distinct from) that for multiple zeta values. We also exploit the Hopf algebra structure of the quasi-symm...
متن کاملRooted Trees and Symmetric Functions: Zhao’s Homomorphism and the Commutative Hexagon
Recent work in perturbative quantum field theory has led to much study of the Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf algebra of rooted trees, had already been studied by algebraists. L. Foissy introduced a noncommutative version of the Connes-Kreimer Hopf algebra, which turns out to be self-dual. Using some homomorphisms defined by the author and W. Zhao, we de...
متن کاملar X iv : m at h / 00 10 14 0 v 1 [ m at h . Q A ] 1 3 O ct 2 00 0 RELATIONS OF MULTIPLE ZETA VALUES AND THEIR ALGEBRAIC EXPRESSION
We establish a new class of relations among the multiple zeta values ζ(k1, . . . , kl) = ∑ n1>···>nl≥1 1 n k1 1 · · ·n kl k , which we call the cyclic sum identities. These identities have an elementary proof, and imply the “sum theorem” for multiple zeta values. They also have a succinct statement in terms of “cyclic derivations” as introduced by Rota, Sagan and Stein. In addition, we discuss ...
متن کامل