Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762



A Pieri-Chevalley formula in the K-theory of a $G/B$-bundle

Authors: Harsh Pittie and Arun Ram
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 102-107
MSC (1991): Primary 14M15; Secondary 14C35, 19E08
Published electronically: July 14, 1999
MathSciNet review: 1701888
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a semisimple complex Lie group, $B$ a Borel subgroup, and $T\subseteq B$ a maximal torus of $G$. The projective variety $G/B$ is a generalization of the classical flag variety. The structure sheaves of the Schubert subvarieties form a basis of the K-theory $K(G/B)$ and every character of $T$ gives rise to a line bundle on $G/B$. This note gives a formula for the product of a dominant line bundle and a Schubert class in $K(G/B)$. This result generalizes a formula of Chevalley which computes an analogous product in cohomology. The new formula applies to the relative case, the K-theory of a $G/B$-bundle over a smooth base $X$, and is presented in this generality. In this setting the new formula is a generalization of recent $G=GL_n({\mathbb C})$ results of Fulton and Lascoux.

References [Enhancements On Off] (What's this?)

  • [Ch] C. Chevalley, Sur les decompositions cellulaires des espaces $G/B$, in Algebraic Groups and their Generalizations: Classical Methods, W. Haboush and B. Parshall eds., Proc. Symp. Pure Math., Vol. 56 Pt. 1, Amer. Math. Soc. (1994), 1-23. MR 95e:14041
  • [FL] W. Fulton and A. Lascoux, A Pieri formula in the Grothendieck ring of a flag bundle, Duke Math. J. 76 (1994), 711-729. MR 96j:14036
  • [FP] W. Fulton and P. Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Math. 1689, Springer-Verlag, Berlin 1998. CMP 98:17
  • [KK] B. Kostant and S. Kumar, $T$-equivariant K-theory of generalized flag varieties, J. Differential Geom. 32 (1990), 549-603. MR 92c:19006
  • [Li] P. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994), 329-346. MR 95f:17023
  • [P] H. Pittie, Homogeneous vector bundles over homogeneous spaces, Topology 11 (1972), 199-203. MR 44:7583
  • [S] R. Steinberg, On a theorem of Pittie, Topology 14 (1975), 173-177. MR 51:9101

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (1991): 14M15, 14C35, 19E08

Retrieve articles in all journals with MSC (1991): 14M15, 14C35, 19E08

Additional Information

Harsh Pittie
Affiliation: Department of Mathematics, Graduate Center, City University of New York, New York, NY 10036

Arun Ram
Affiliation: Department of Mathematics, Princeton University, Princeton, NJ 08544

Received by editor(s): February 9, 1999
Published electronically: July 14, 1999
Additional Notes: Research supported in part by National Science Foundation grant DMS-9622985.
Communicated by: Efim Zelmanov
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society