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Abstract In this paper, we give Pieri rules for skew dual immaculate functions and their recently discovered row-strict counterparts. We establish our rules using a right-action analogue of the skew Littlewood–Richardson rule for Hopf algebras of Lam–Lauve–Sottile. We also obtain Pieri rules for row-strict (dual) immaculate functions. 

Abstract Working in a polynomial ring , where is an arbitrary commutative ring with 1, we consider the th Veronese subalgebras , as well as natural ‐submodules inside . We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple ‐equivariant minimal free ‐resolutions for the quotient ring and for these modules . These also lead to elegant descriptions of for all and for any pair of these modules . 

We introduce a new basis of quasisymmetric functions, the row-strict dual immaculate functions. We construct a cyclic, indecomposable 0-Hecke algebra module for these functions. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution $\mathrm{\psi <\#comment/>}$on the ring $\mathrm{QSym}$of quasisymmetric functions. We give an explicit description of the effect of $\mathrm{\psi <\#comment/>}$on the associated 0-Hecke modules, via the poset induced by the 0-Hecke action on standard immaculate tableaux. This remarkable poset reveals other 0-Hecke submodules and quotient modules, often cyclic and indecomposable, notably for a row-strict analogue of the extended Schur functions studied in Assaf-Searles (2019). Like the dual immaculate function, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, corresponding to a specific descent set. We give a complete combinatorial and representation-theoretic picture by constructing 0-Hecke modules for the remaining variations on descent sets, and showing thatallthe possible variations for generating functions of tableaux occur as characteristics of the 0-Hecke modules determined by these descent sets. 

We give bijective proofs of Monk's rule for Schubert and double Schubert polynomials computed with bumpless pipe dreams. In particular, they specialize to bijective proofs of transition and cotransition formulas of Schubert and double Schubert polynomials, which can be used to establish bijections with ordinary pipe dreams.