Noncommutative irreducible characters of the symmetric group and noncommutative Schur functions

TL;DR

This paper demonstrates that in the noncommutative symmetric functions algebra, certain bases of noncommutative Schur functions correspond to noncommutative irreducible characters of symmetric groups, extending classical representation theory.

Contribution

It establishes the isomorphism between noncommutative Schur functions and noncommutative irreducible characters, revealing new algebraic structures in noncommutative character theory.

Findings

Noncommutative Schur functions correspond to noncommutative irreducible characters.

Young noncommutative Schur functions also correspond to irreducible characters.

The results extend classical symmetric function theory to noncommutative settings.

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.