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 satises that every element is isomorphic to a noncommutative irreducible character of a symmetric group.