A Hopf algebra generalization of the symmetric functions in partially commutative variables

The quasisymmetric functions, $QSym$, are generalized for a finite alphabet $A$ by the colored quasisymmetric functions, $QSym_A$, in partially commutative variables. Their dual, $NSym_A$, generalizes the noncommutative symmetric functions, $NSym$, through a relationship with a Hopf algebra of trees. We define an algebra $Sym_A$, contained within $QSym_A$, that is isomorphic to the symmetric functions, $Sym$, when $A$ is an alphabet of size one. We show that $Sym_A$ is a Hopf algebra and define its graded dual, $PSym_A$, which is the commutative image of $NSym_A$ and also generalizes $Sym$. The seven algebras listed here can be placed in a commutative diagram connected by Hopf morphisms. In addition to defining generalizations of the classic bases of the symmetric functions to $Sym_A$ and $PSym_A$, we describe multiplication, comultiplication, and the antipode in terms of a basis for both algebras. We conclude by defining a pair of dual bases that generalize the Schur functions and listing open questions.