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

Kumar described the Schubert classes which are the dual to the closures of the Bruhat cells in the flag varieties of the Kac-Moody groups associated to the infinite dimensional KacMoody algebras [17]. These classes are indexed by affine Weyl groups and can be chosen as elements of integral cohomologies of the homogeneous space L̂polGC/B̂ for any compact simply connected semisimple Lie groupG. Later, S. Kumar, and B. Kostant described explicit cup product formulas of these classes in the cohomology algebras by using the relation between the invariant-theoretic relative Lie algebra cohomology theory (using the representation module of the nilpotent part) with the purely nil-Hecke rings [16]. These explicit product formulas involve some BGG-type operators Ai and reflections. In the published work [20] of the first author, using some homotopy equivalences, cohomology ring structures of LG/T have been determined where LG is the smooth loop space on G. He has calculated the products and explicit ring structure of LSU2/T using these ideas. He found that it has a quotient of the divided power algebra. In this work, we list explicit presentation of affine Weyl group of the loop group LSU3. We calculate generators for ideals and the rank of the modules of graded cohomology algebra of LSU3/T and ΩSU3 in the coefficient ring Z[1/2]. Some comments about the structure of this work are in order. It is written for a reader with a first course in algebraic topology and some understanding of the structure of