Extended Schur functions and bases related by involutions

Abstract

We introduce two new bases of QSym, the flipped extended Schur functions and the backward extended Schur functions, as well as their duals in NSym, the flipped shin functions and the backward shin functions. These bases are the images 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 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 $\rho$ and $\omega$ to translate these and other known results to our flipped and backward bases.

28 pages

References in corpus (11)

• Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
• Hopf Algebras in Combinatorics
• Quasisymmetric Schur functions
• Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables
• A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions
• Multiplicative structures of the immaculate basis of non-commutative symmetric functions
• The Pieri rule for dual immaculate quasi-symmetric functions
• Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables
• 0-Hecke modules for row-strict dual immaculate functions
• Rigidity for the Hopf algebra of quasi-symmetric functions
• Pieri rules for skew dual immaculate functions