On the Hopf Algebra of Noncommutative Symmetric Functions in Superspace

We study the combinatorial Hopf algebra of noncommutative symmetric functions in superspace sNSym, introduced by Fishel, Lapointe and Pinto. We introduce a family of primitive elements of sNSym and extend the noncommutative elementary and power sum functions to superspace. Then, we give formulas relating these families of functions. Also, we introduce noncommutative Ribbon Schur functions in superspace and provide an explicit formula for their product. We show that the dual basis of these functions is given by a family of the so–called fundamental quasisymmetric functions in superspace. This allows us to obtain an explicit formula for the coproduct of fundamental quasisymmetric functions in superspace. Additionally, by projecting the noncommutative Ribbon Schur functions in superspace, we define a new basis for the algebra of symmetric functions in superspace.