International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 86494, 55 pages
doi:10.1155/IJMMS/2006/86494
Combinatorial integers (m,nj) and Schubert calculus in the integral cohomology ring of infinite smooth flag manifolds
Department of Mathematics, Abant Izzet Baysal University (AIBU), Golkoy Campus, Bolu 14280, Turkey
Received 18 July 2005; Revised 22 February 2006; Accepted 25 April 2006
Copyright © 2006 Cenap Özel and Erol Yilmaz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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].