Schubert calculus in complex cobordism

Bressler, Paul; Evens, Sam

doi:10.1090/S0002-9947-1992-1044959-0

Schubert calculus in complex cobordism
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Trans. Amer. Math. Soc. 331 (1992), 799-813 Request permission

Abstract:

We study the structure of the complex cobordism ring of the flag variety of a compact connected Lie group. An explicit procedure for determining products of basis elements is obtained, generalizing the work of Bernstein-Gel’fand-Gel’fand on ordinary cohomology and of Kostant-Kumar on $K$-theory. Bott-Samelson resolutions are used to replace the classical basis of Schubert cells.

References

Alberto Arabia, Cohomologie $T$-équivariante de la variété de drapeaux d’un groupe de Kac-Moody, Bull. Soc. Math. France 117 (1989), no. 2, 129–165 (French, with English summary). MR 1015806
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Schubert cells, and the cohomology of the spaces $G/P$, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3–26 (Russian). MR 0429933
Sam Evens and Paul Bressler, On certain Hecke rings, Proc. Nat. Acad. Sci. U.S.A. 84 (1987), no. 3, 624–625. MR 873070, DOI 10.1073/pnas.84.3.624
Paul Bressler and Sam Evens, The Schubert calculus, braid relations, and generalized cohomology, Trans. Amer. Math. Soc. 317 (1990), no. 2, 799–811. MR 968883, DOI 10.1090/S0002-9947-1990-0968883-2
A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 102800, DOI 10.2307/2372795
Raoul Bott and Hans Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029. MR 105694, DOI 10.2307/2372843
Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). MR 354697
Eldon Dyer, Cohomology theories, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0268883
Eugene Gutkin, Representations of Hecke algebras, Trans. Amer. Math. Soc. 309 (1988), no. 1, 269–277. MR 957070, DOI 10.1090/S0002-9947-1988-0957070-0
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
Victor G. Kac, Constructing groups associated to infinite-dimensional Lie algebras, Infinite-dimensional groups with applications (Berkeley, Calif., 1984) Math. Sci. Res. Inst. Publ., vol. 4, Springer, New York, 1985, pp. 167–216. MR 823320, DOI 10.1007/978-1-4612-1104-4_{7}
Bertram Kostant and Shrawan Kumar, The nil Hecke ring and cohomology of $G/P$ for a Kac-Moody group $G$, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 6, 1543–1545. MR 831908, DOI 10.1073/pnas.83.6.1543
Bertram Kostant and Shrawan Kumar, $T$-equivariant $K$-theory of generalized flag varieties, Proc. Nat. Acad. Sci. U.S.A. 84 (1987), no. 13, 4351–4354. MR 894705, DOI 10.1073/pnas.84.13.4351
Shrawan Kumar, Demazure character formula in arbitrary Kac-Moody setting, Invent. Math. 89 (1987), no. 2, 395–423. MR 894387, DOI 10.1007/BF01389086
Daniel Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971), 29–56 (1971). MR 290382, DOI 10.1016/0001-8708(71)90041-7
Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Mathematical notes. MR 0248858