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

Published electronically:
July 14, 1999

MathSciNet review:
1701888

Full-text PDF Free Access

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 décompositions cellulaires des espaces 𝐺/𝐵*, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 1–23 (French). With a foreword by Armand Borel. MR**1278698****[FL]**William Fulton and Alain Lascoux,*A Pieri formula in the Grothendieck ring of a flag bundle*, Duke Math. J.**76**(1994), no. 3, 711–729. MR**1309327**, 10.1215/S0012-7094-94-07627-8**[FP]**W. Fulton and P. Pragacz,*Schubert varieties and degeneracy loci*, Lecture Notes in Math.**1689**, Springer-Verlag, Berlin 1998. CMP**98:17****[KK]**Bertram Kostant and Shrawan Kumar,*𝑇-equivariant 𝐾-theory of generalized flag varieties*, J. Differential Geom.**32**(1990), no. 2, 549–603. MR**1072919****[Li]**Peter Littelmann,*A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras*, Invent. Math.**116**(1994), no. 1-3, 329–346. MR**1253196**, 10.1007/BF01231564**[P]**Harsh V. Pittie,*Homogeneous vector bundles on homogeneous spaces*, Topology**11**(1972), 199–203. MR**0290402****[S]**Robert Steinberg,*On a theorem of Pittie*, Topology**14**(1975), 173–177. MR**0372897**

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

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