A Pieri-Chevalley formula in the K-theory of a -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

DOI:
https://doi.org/10.1090/S1079-6762-99-00067-0

Published electronically:
July 14, 1999

MathSciNet review:
1701888

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a semisimple complex Lie group, a Borel subgroup, and a maximal torus of . The projective variety is a generalization of the classical flag variety. The structure sheaves of the Schubert subvarieties form a basis of the K-theory and every character of gives rise to a line bundle on . This note gives a formula for the product of a dominant line bundle and a Schubert class in . 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 -bundle over a smooth base , and is presented in this generality. In this setting the new formula is a generalization of recent results of Fulton and Lascoux.

**[Ch]**C. Chevalley,*Sur les decompositions cellulaires des espaces*, 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,*-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**

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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

Email:
rama@math.princeton.edu

DOI:
https://doi.org/10.1090/S1079-6762-99-00067-0

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