The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. As the power sums generate the algebra of symmetric functions, the Murnaghan-Nakayama rule is as fundamental as the Pieri rule. Interesting, the resulting formulas from the Murnaghan-Nakayama rule are significantly more compact than those from the Pieri formula. In geometry, a Murnaghan-Nakayama formula computes the intersection of Schubert cycles with tautological classes coming from the Chern character.
In this talk, I will discuss some background, and then some recent work with Andrew Morrison establishing Murnaghan-Nakayama rules for Schubert polynomials and for the quantum cohomology of the Grassmannian. The results I discuss are contained in the preprint arXiv:1507.06569.