{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:35:49Z","timestamp":1753886149998,"version":"3.41.2"},"reference-count":23,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,8,31]],"date-time":"2006-08-31T00:00:00Z","timestamp":1156982400000},"content-version":"vor","delay-in-days":242,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>We discuss the calculation of integral cohomology ring of <jats:italic>L<\/jats:italic><jats:italic>G<\/jats:italic>\/<jats:italic>T<\/jats:italic> and <jats:italic>\u03a9<\/jats:italic><jats:italic>G<\/jats:italic>. First we describe the root system and Weyl group of <jats:italic>L<\/jats:italic><jats:italic>G<\/jats:italic>, then we give some homotopy equivalences on the loop groups and homogeneous spaces, and calculate the cohomology ring structures of <jats:italic>L<\/jats:italic><jats:italic>G<\/jats:italic>\/<jats:italic>T<\/jats:italic> and <jats:italic>\u03a9<\/jats:italic><jats:italic>G<\/jats:italic> for affine group . We introduce combinatorial integers  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 <jats:italic>\u2124<\/jats:italic>[1\/2].<\/jats:p>","DOI":"10.1155\/ijmms\/2006\/86494","type":"journal-article","created":{"date-parts":[[2006,9,28]],"date-time":"2006-09-28T07:25:22Z","timestamp":1159428322000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Combinatorial integers (m,nj) and Schubert calculus in the integral cohomology ring of infinite smooth flag manifolds"],"prefix":"10.1155","volume":"2006","author":[{"given":"Cenap","family":"\u00d6zel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Erol","family":"Yilmaz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,8,31]]},"reference":[{"key":"e_1_2_1_1_2","series-title":"Actualit\u00e9s Scientifiques et Industrielles, no. 1337","volume-title":"\u00c9l\u00e9ments de math\u00e9matique. Fasc. XXXIV. Groupes et alg\u00e8bres de Lie. Chapitre IV: Groupes de Coxeter et syst\u00e8mes de Tits. Chapitre V: Groupes engendr\u00e9s par des r\u00e9flexions. Chapitre VI: syst\u00e8mes de racines","author":"Bourbaki N.","year":"1968"},{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-6848-0"},{"volume-title":"Lectures on Analysis. Vol. I: Integration and Topological Vector Spaces","year":"1969","author":"Choquet G.","key":"e_1_2_1_3_2"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2693-0"},{"key":"e_1_2_1_5_2","series-title":"Graduate Texts in Mathematics","volume-title":"Using Algebraic Geometry","author":"Cox D. A.","year":"2005"},{"key":"e_1_2_1_6_2","doi-asserted-by":"crossref","unstructured":"GabberO.andKacV. G. 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