Quasisymmetric functions from a topological point of view

By Andrew H. Baker, Birgit Richter. 2006

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It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions $\mathsf {Symm}$. We offer the cohomology of the space $\Omega \Sigma {\mathsf C} P^{\infty}$ as a topological model for the ring of quasisymmetric functions $\mathsf {QSymm}$. We exploit standard results from topology to shed light on some of the algebraic properties of $\mathsf {QSymm}$. In particular, we reprove the Ditters conjecture. We investigate a product on $\Omega \Sigma {\mathsf C} P^{\infty}$ that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of $BU$. The canonical Thom spectrum over $\Omega \Sigma {\mathsf C} P^{\infty}$ is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e8b2260831b178cf95ace71b818eed93727c088b

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In Discrete Mathematics and Combinatorics

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