Diagrams and reduced decompositions for cominuscule flag varieties and affine Grassmannians.

We develop a system of canonical reduced decompositions of minimal coset representatives of quotients corresponding to cominuscule flag varieties and affine Grassmannians. This canonical decomposition allows, in the first case, an abbreviated computation of relative R-polynomials. From this, we show that these polynomials can be obtained from unlabelled intervals, and more generally, that Kazhdan-Lusztig polynomials associated to cominuscule flag varieties are combinatorially invariant. In the second case, we are able to provide a list of the rationally smooth Schubert varieties in simply laced affine Grassmannians corresponding to types A, D, and E. The results in this case were obtained independently by Billey and Mitchell in 2008.