Kostant polynomials and the cohomology ring for G/B

Kostant polynomials and the cohomology ring for G/B

Sara C. Billey†

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139

Abstract

The Schubert calculus for G/B can be completely determined by a certain matrix related to the Kostant polynomials introduced in section 5 of Bernstein, Gelfand, and Gelfand [Bernstein, I., Gelfand, I. & Gelfand, S. (1973) Russ. Math. Surv. 28, 1–26]. The polynomials are defined by vanishing properties on the orbit of a regular point under the action of the Weyl group. For each element w in the Weyl group the polynomials also have nonzero values on the orbit points corresponding to elements which are larger in the Bruhat order than w. The main theorem given here is an explicit formula for these values. The matrix of orbit values can be used to determine the cup product for the cohomology ring for G/B, using only linear algebra or as described by Lascoux and Schützenberger [Lascoux, A. & Schützenberger, M.-P. (1982) C. R. Seances Acad. Sci. Ser. A 294, 447–450]. Complete proofs of all the theorems will appear in a forthcoming paper.

Footnotes

↵ † To whom reprint requests should be addressed. e-mail: billey{at}math.mit.edu.
Bertram Kostant, Massachusetts Institute of Technology, Cambridge, MA
↵ ‡ This notation differs from the notation in ref. 1 by a sign. The result is that we have interchanged the positive and the negative roots from those used in ref. 1.
↵ § Note that our d _uv corresponds with $\text{[math]}$ in ref. 2.