Extended Schur functions and bases related By involutions

We introduce two new bases of QSym, the flipped extended Schur functions andthe backward extended Schur functions, as well as their duals in NSym, theflipped shin functions and the backward shin functions. These bases are theimages of the extended Schur basis and shin basis under the involutions $\rho$and $\omega$ on the quasisymmetric and noncommutative symmetric functions,which generalize the classical involution $\omega$ on the symmetric functions.In addition, we prove a Jacobi-Trudi rule for certain shin functions usingcreation operators. We define skew extended Schur functions and skew-IIextended Schur functions based on left and right actions of NSym and QSymrespectively. We then use the involutions $\rho$ and $\omega$ to translatethese and other known results to our flipped and backward bases.