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The algebra of quasisymmetric functions QSym and the shuffle algebra ofcompositions Sh are isomorphic as graded Hopf algebras (in characteristiczero), and isomorphisms between them can be specified via shuffle bases ofQSym. We use the notion of infinitesimal characters to characterize shufflebases, and we establish a universal property for Sh in the category ofconnected graded Hopf algebras equipped with an infinitesimal character,analogous to the universal property of QSym as a combinatorial Hopf algebradescribed by Aguiar, Bergeron, and Sottile. We then use these results to givegeneral constructions for quasisymmetric power sums, recovering four previousconstructions from the literature, and study their properties.

Shuffle Bases and Quasisymmetric Power Sums