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The Schubert calculus, braid relations, and generalized cohomology


Authors: Paul Bressler and Sam Evens
Journal: Trans. Amer. Math. Soc. 317 (1990), 799-811
MSC: Primary 57T15; Secondary 22E45, 32M10, 55N20
DOI: https://doi.org/10.1090/S0002-9947-1990-0968883-2
MathSciNet review: 968883
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Abstract: Let $ X$ be the flag variety of a compact Lie group and let $ {h^{\ast}}$ be a complex-oriented generalized cohomology theory. We introduce operators on $ {h^{\ast}}(X)$ which generalize operators introduced by Bernstein, Gel'fand, and Gel'fand for rational cohomology and by Demazure for $ K$-theory. Using the Becker-Gottlieb transfer, we give a formula for these operators, which enables us to prove that they satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by the authors and extended by Gutkin.


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  • [1] J. F. Adams, Stable homotopy and generalised homology, Univ. of Chicago Press, 1974. MR 0402720 (53:6534)
  • [2] E. Akyildiz and J. B. Carrell, Zeros of holomorphic vector fields and the Gysin homomorphism, Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, R.I., 1983, pp. 47-54. MR 713044 (85c:32020)
  • [3] E. Artin, Galois theory, Notre Dame lecture notes, 1959. MR 0265324 (42:234)
  • [4] M. F. Atiyah and R. Bott, A Lefschetz fixed point theorem for elliptic complexes. II, Ann. of Math. 88 (1968), 451-491. MR 0232406 (38:731)
  • [5] M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math., vol. 3, Amer. Math. Soc., Providence, R.I., 1961, pp. 7-38. MR 0139181 (25:2617)
  • [6] J. C. Becker and D. H. Gottlieb, The transfer map and fiber bundles, Topology 14 (1975), 1-12. MR 0377873 (51:14042)
  • [7] J. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand, Schubert cells and the cohomology of spaces $ G/P$, Russian Math Surveys 28 (1973), 1-26.
  • [8] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458-538. MR 0102800 (21:1586)
  • [9] N. Bourbaki, Groupes et algèbres de Lie, Hermann, Paris, 1968, Chapitres 4, 5, et 6. MR 0240238 (39:1590)
  • [10] P. Bressler and S. Evens, On certain Hecke rings, Proc. Nat. Acad. Sci. U.S.A. 84 (1987), 624-625. MR 873070 (88e:17011)
  • [11] G. Brumfiel and I. Madsen, Evaluation of the transfer and universal surgery classes, Invent. Math. 32 (1976), 133-169. MR 0413099 (54:1220)
  • [12] M. Demazure, Désingularisation des variétés de Schubert, Ann. Ecole Norm. Sup. (4) 7 (1974), 53-88. MR 0354697 (50:7174)
  • [13] E. Dyer, Cohomology theories, Benjamin, 1969. MR 0268883 (42:3780)
  • [14] S. Evens, The transfer for compact Lie groups, induced representations, and braid relations, Thesis MIT, 1988.
  • [15] E. Gutkin, Representations of Hecke algebras, Trans. Amer. Math. Soc. 307, 1988. MR 957070 (90c:22030)
  • [16] J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, 1972. MR 0323842 (48:2197)
  • [17] V. Kač, Constructing groups associated to infinite dimensional Lie algebras, Infinite Dimensional Groups with Applications, Springer-Verlag, 1985, pp. 167-217. MR 823320 (87c:17024)
  • [18] D. Kazhdan and G. Lusztig, Equivariant $ K$-theory and representation of Hecke algebras. II, Invent. Math. 80 (1985), 209-231. MR 788408 (88f:22054b)
  • [19] B. Kostant and S. Kumar, The nil Hecke ring and cohomology of $ G/P$ for a Kač-Moody group $ G$, Adv. in Math. 62 (1986), 187-237. MR 866159 (88b:17025b)
  • [20] B. Kostant and S. Kumar, $ T$-equivariant $ K$-theory of generalized flag varieties, Proc. Nat. Acad. Sci. U.S.A. 84 (1987), 4351-4354. MR 894705 (88m:22048)
  • [21] G. Lusztig, Equivariant $ K$-theory and representations of Hecke algebras, Proc. Amer. Math. Soc. 94 (1985), 337-342. MR 784189 (88f:22054a)
  • [22] S. A. Mitchell and S. B. Priddy, A double coset formula for Levi subgroups and splitting $ BG{L_n}$, preprint, 1987. MR 710102 (85f:55005)
  • [23] D. G. Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Adv. in Math. 7 (1971), 29-56. MR 0290382 (44:7566)
  • [24] M. F. Atiyah and F. Hirzebruch, The Riemann-Roch theorem for analytic embeddings, Topology 1 (1962), 151-166. MR 0148084 (26:5593)
  • [25] M. Hazewinkel, Formal groups and applications, Academic Press, 1978. MR 506881 (82a:14020)

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DOI: https://doi.org/10.1090/S0002-9947-1990-0968883-2
Article copyright: © Copyright 1990 American Mathematical Society

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