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We discuss the calculation of integral cohomology ring of LG/T and ΩG. First we describe the root system and Weyl group of LG, then we give some homotopy equivalences on the loop groups and homogeneous spaces, and calculate the cohomology ring structures of LG/T and ΩG for affine group A^2. We introduce combinatorial integers (m,nj) 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]

international journal of mathematics and mathematical sciences

Combinatorial integers<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="E1"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>j</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>and Schubert calculus in the integral cohomology ring of infinite smooth flag manifolds

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