Benedetti, CarolinaBergeron, NantelColmenarejo, LauraSaliola, FrancoSottile, Frank2025-04-112025-04-112024-06-08https://doi.org/10.48550/arXiv.2406.05311https://hdl.handle.net/10315/4289434 pages, 16 color pictures, some text and graphic in color. Full paper with all proofsIn this paper, we give a rule for the multiplication of a Schubert class by a tautological class in the (small) quantum cohomology ring of the flag manifold. As an intermediate step, we establish a formula for the multiplication of a Schubert class by a quantum Schur polynomial indexed by a hook partition. This entails a detailed analysis of chains and intervals in the quantum Bruhat order. This analysis allows us to use results of Leung--Li and of Postnikov to reduce quantum products by hook Schur polynomials to the (known) classical product.enAttribution-NonCommercial-ShareAlike 4.0 InternationalCombinatoricsAlgebraic geometryQuantum algebra05E0514N15Preprint