The primitives of the Hopf algebra of noncommutative symmetric functions

11/2004; DOI: 10.11606/issn.2316-9028.v1i2p175-202
Source: arXiv


Let NSymm be the Hopf algebra of noncommutative symmetric functions over the integers. In this paper a description is given of its Lie algebra of primitives over the integers, Prim(NSymm), in terms of recursion formulas. For each of the primitives of a basis of Prim(NSymm), indexed by Lyndon words, there is a recursively given divided power series over it. This gives another proof of the theorem that the algebra of quasi-symmetric functions is free over the integers.

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Available from: Michiel Hazewinkel, Apr 28, 2015
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    • "The algebras NSymm and QSymm have received a great deal of attention in combinatorics . Several structural properties were proven, for instance about the explicit form of the primitives in the coalgebra NSymm [19] or the freeness of QSymm as a commutative algebra [16]. The latter result is known as the Ditters conjecture, and is our Theorem 2.1. "
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    ABSTRACT: It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions, Symm. We offer the cohomology of the loop space of the suspension of the infinite complex projective space as a topological model for the ring of quasisymmetric functions, QSymm. We exploit standard results from topology to shed light on some of the algebraic properties of QSymm. In particular, we reprove the Ditters conjecture. We investigate a product on the loop space that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of BU. The canonical Thom spectrum over the loops on the suspension of BU(1) is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.
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    ABSTRACT: We introduce certain set W of quasisymmetric functions, called Lyndon-Witt functions, since they are parametrized by Lyndon compositions and are in the same time ghost components of global Witt vectors over QSym . Using results from [DS], we correct an error, noted independently by M. Hazewinkel and Chr. Reutenauer, and show thatQSym = Z[W], a free polynomial ring. We present a counterexample to the main theorem of [DS]: the monomial quasisymmetric function indexed by the composition (3, 6) is in Z[L mod , 1 7 ] and has not all its coefficients in Z. Author's email adress is A site (in preparation) is∼ejd/qslw.
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