Schubert calculus in complex cobordism
HTML articles powered by AMS MathViewer
- by Paul Bressler and Sam Evens PDF
- 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
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 331 (1992), 799-813
- MSC: Primary 57R77
- DOI: https://doi.org/10.1090/S0002-9947-1992-1044959-0
- MathSciNet review: 1044959