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.