Murnaghan-Nakayama Rules in Schubert Calculus

   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.