Extended Schur functions and bases related by involutions

Abstract

We introduce two new bases of $QSym$, the reverse extended Schur functions and the row-strict reverse extended Schur functions, as well as their duals in NSym, the reverse shin functions and row-strict reverse shin functions. These bases are the images of the extended Schur basis and shin basis under the involutions ρ and ω, which generalize the classical involution ω on the symmetric functions. In addition, we prove a Jacobi-Trudi rule for certain shin functions using creation operators. We define skew extended Schur functions and skew-II extended Schur functions based on left and right actions of NSym and $QSym$ respectively. We then use the involutions ρ and ω to translate these and other known results to our reverse and row-strict reverse bases.