Combinatorial integers ( m , n j ) and Schubert calculus in the integral cohomology ring of infinite smooth flag manifolds

We discuss the calculation of integral cohomology ring of L G / T and Ω G . First we describe the root system and Weyl group of L G , then we give some homotopy equivalences on the loop groups and homogeneous spaces, and calculate the cohomology ring structures of L G / T and Ω G for affine group A ^ 2 . We introduce combinatorial integers ( m , n j ) which play a crucial role in our calculations and give some interesting identities among these integers. Last we calculate generators for ideals and rank of each module of graded integral cohomology algebra in the local coefficient ring ℤ [ 1 / 2 ] .