The Schubert calculus, braid relations, and generalized cohomology

Bressler, Paul; Evens, Sam

doi:10.1090/S0002-9947-1990-0968883-2

The Schubert calculus, braid relations, and generalized cohomology
HTML articles powered by AMS MathViewer

by Paul Bressler and Sam Evens PDF

Trans. Amer. Math. Soc. 317 (1990), 799-811 Request permission

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.

References

J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0402720
E. Akyıldız and J. B. Carrell, Zeros of holomorphic vector fields and the Gysin homomorphism, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 47–54. MR 713044
Emil Artin, Galois theory, Notre Dame Mathematical Lectures, no. 2, University of Notre Dame Press, South Bend, Ind., 1959. Edited and supplemented with a section on applications by Arthur N. Milgram; Second edition, with additions and revisions; Fifth reprinting. MR 0265324
M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451–491. MR 232406, DOI 10.2307/1970721
M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 7–38. MR 0139181
J. C. Becker and D. H. Gottlieb, The transfer map and fiber bundles, Topology 14 (1975), 1–12. MR 377873, DOI 10.1016/0040-9383(75)90029-4

Schubert cells and the cohomology of spaces

28

A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 102800, DOI 10.2307/2372795
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
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
G. Brumfiel and I. Madsen, Evaluation of the transfer and the universal surgery classes, Invent. Math. 32 (1976), no. 2, 133–169. MR 413099, DOI 10.1007/BF01389959
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

The transfer for compact Lie groups, induced representations, and braid relations

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
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
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}
George Lusztig, Equivariant $K$-theory and representations of Hecke algebras, Proc. Amer. Math. Soc. 94 (1985), no. 2, 337–342. MR 784189, DOI 10.1090/S0002-9939-1985-0784189-2
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
George Lusztig, Equivariant $K$-theory and representations of Hecke algebras, Proc. Amer. Math. Soc. 94 (1985), no. 2, 337–342. MR 784189, DOI 10.1090/S0002-9939-1985-0784189-2
Stephen A. Mitchell and Stewart B. Priddy, Stable splittings derived from the Steinberg module, Topology 22 (1983), no. 3, 285–298. MR 710102, DOI 10.1016/0040-9383(83)90014-9
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
M. F. Atiyah and F. Hirzebruch, The Riemann-Roch theorem for analytic embeddings, Topology 1 (1962), 151–166. MR 148084, DOI 10.1016/0040-9383(65)90023-6
Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506881