  Some new relations on skew Schur function differences are established both
combinatorially using Sch\"utzenberger's jeu de taquin, and algebraically using
Jacobi-Trudi determinants. These relations lead to the conclusion that certain
differences of skew Schur functions are Schur positive. Applying these results
to a basis of symmetric functions involving ribbon Schur functions confirms the
validity of a Schur positivity conjecture due to McNamara. A further
application reveals that certain differences of products of Schubert classes
are Schubert positive.
