Noncommutative irreducible characters of the symmetric group and noncommutative Schur functions

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.

Keywords

descent algebra, irreducible character, noncommutative character theory, noncommutative symmetric function, Schur function, symmetric group

2010 Mathematics Subject Classification

Primary 05E05, 16T30. Secondary 05E10, 16T05, 20B30, 30C30, 33D52.

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Published 27 December 2013