A(nother) topologist ponders the ring of quasi-symmetric functions

Nick Kuhn (University of Virginia)

Wednesday 22nd February, 2017 15:00-16:00 Maths 522


The ring quasi-symmetric functions, QSymm, is an analogue of the ring of symmetric functions.  This ring appears as the cohomology ring of a topological space familiar to algebraic topologists.   In 2008, Andy Baker (at Glasgow) and Birget Richter used this observations, together with well known topological decomposition results to give a very elementary proof of Hazewinkel's theorem that QSymm is a polynomial algebra.

I revisit and reorganize their ideas, in the process eliminating their apparent dependence on algebraic topology.  More generally, I construct an interesting functor from graded commutative algebras to Hopf algebras, that outputs QSymm when one inputs  a polynomial algebra on one generator.  Topological considerations suggest algebraic constructions, e.g., a Hopf algebra embedding of QSymm into the Shuffle algebra that may be new.

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