Three Examples of Quasisymmetric Compatible Sₙ-modules

The Schur functions, a basis for the symmetric polynomials (Sym), encode the irreducible representations of the symmetric group, Sₙ, via the Frobenius characteristic map. In 1996, Krob and Thibon defined a quasisymmetric Frobenius map on the representations of Hₙ (0), mapping them to the quasisymmetric functions (QSym). Despite the obvious inclusion of Sym in QSym and the close relationship between Sₙ and Hₙ (0), there is no known direct link between these two Frobenius characteristic maps and the related representations. We explore three specific situations in which a deformation of an Sₙ action results in a valid Hₙ (0) action and gives a quasisymmetric Frobenius characteristic that is equal to the symmetric Frobenius characteristic. We introduce the concept of quasisymmetric compatibility, which formalizes a link between the two maps, and we show it applies to all Sₙ-modules.