Noncommutative irreducible characters of the symmetric group and noncommutative Schur functions

Abstract: In the Hopf algebra of symmetric functions, Sym, the basis of Schur functions is distinguished since every Schur function is isomorphic to an irreducible character of a symmetric group under the Frobenius characteristic map. In this note we show that in the Hopf algebra of noncommutative symmetric functions, Nsym, of which Sym is a quotient, the recently discovered basis of noncommutative Schur functions exhibits that every noncommutative Schur function is isomorphic to a noncommutative irreducible character of a symmetric group when working in noncommutative character theory. We simultaneously show that a second basis of Nsym consisting of Young noncommutative Schur functions also satisfies that every element is isomorphic to a noncommutative irreducible character of a symmetric group.