The Cartan matrix and enumerative calculus

Let G be a compact connected Lie group and P G be the centralizer of a one-parameter subgroup in G. We explain a program that reduces integration along a Schubert variety in the flag manifold G/P to the Cartan matrix of G. As applications of the program, we complete the project of explicit computation of the degree and Chern number of an arbitrary Schubert variety started in Zhao and Duan (J. Symbolic Comput. 33 (2002) 507).