Lifting the dual immaculate functions

Abstract

We introduce a reverse variant of the dual immaculate quasisymmetric functions, mirroring the dichotomy between quasisymmetric Schur functions and Young quasisymmetric Schur functions, and establish a lift of this basis to the polynomial ring. We show that taking stable limits of these reverse dual immaculate slide polynomials produces the reverse dual immaculate quasisymmetric functions, and we establish positive formulas for expansions of these polynomials into the fundamental slide and quasi-key bases for polynomials. These formulas mirror connections between dual immaculate quasisymmetric functions, fundamental quasisymmetric functions, and Young quasisymmetric Schur functions, extending these connections from the ring of quasisymmetric functions to the full polynomial ring. We also use this lift to provide a lift of the dual immaculate quasisymmetric functions, and we establish analogous expansion formulas. We moreover use the reverse dual immaculate quasisymmetric functions to establish a connection between the dual immaculate quasisymmetric functions and the Demazure atom basis for polynomials.