Equality of Dual Immaculate Functions Under Automorphisms

The dual immaculate functions are an example of a Schur-like basis in the algebra of quasisymmetric functions. We classify when the image of a dual immaculate function under one of the involutions $ρ, ψ, ω$ is equal to a dual immaculate function. As well, the leading term of the transition matrix is identified, and sufficient and necessary conditions for the existence of immaculate tableaux are determined. As a consequence, new maps and canonical tableaux associated with compositions are discovered.