Billey S C
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
Proc Natl Acad Sci U S A. 1997 Jan 7;94(1):29-32. doi: 10.1073/pnas.94.1.29.
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.
G/B的舒伯特演算可以由一个与伯恩斯坦、盖尔范德和盖尔范德[伯恩斯坦，I.，盖尔范德，I. & 盖尔范德，S.（1973年）《俄罗斯数学综述》28卷，1 - 26页]第5节中引入的科斯坦特多项式相关的特定矩阵完全确定。这些多项式由在正则点的外尔群作用轨道上的消失性质定义。对于外尔群中的每个元素w，这些多项式在与布鲁哈特序中比w大的元素相对应的轨道点上也有非零值。这里给出的主要定理是这些值的一个显式公式。轨道值矩阵可用于确定G/B上同调环的上积，只需使用线性代数，或者如拉斯库和舒岑贝格尔[拉斯库，A. & 舒岑贝格尔，M.-P.（1982年）《法国科学院院报，A辑》294卷，447 - 450页]所描述的那样。所有定理的完整证明将在即将发表的一篇论文中出现。
