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We introduce two new bases for polynomials that lift monomial and fundamentalquasisymmetric functions to the full polynomial ring. By defining a newcondition on pipe dreams, called quasi-Yamanouchi, we give a positivecombinatorial rule for expanding Schubert polynomials into these new bases thatparallels the expansion of Schur functions into fundamental quasisymmetricfunctions. As a result, we obtain a refinement of the stable limits of Schubertpolynomials to Stanley symmetric functions. We also give combinatorial rulesfor the positive structure constants of these bases that generalize thequasi-shuffle product and shuffle product, respectively. We use this to give aLittlewood--Richardson rule for expanding a product of Schubert polynomialsinto fundamental slide polynomials and to give formulas for products of Stanleysymmetric functions in terms of Schubert structure constants.

Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams