A Murnaghan-Nakayama formula in quantum 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   establishing  Murnaghan-Nakayama   rules  for
Schubert polynomials  and for quantum  Schubert polynomials.
This  is  joint  work with  Morrison,  Benedetti,  Bergeron,
Ccolmenarejo, and Saliola.