Mathematical Sciences: Schubert Varieties and Standard Monomial Theory

这项研究将集中在与一个 域k上的半单单连通代数群G， G的Borel子群B与旗上的Schubert簇 品种G/B。 私家侦探发展了一个标准的单项理论 作为经典Hodge-Young标准的推广 单项理论 以下具体问题将 考虑了 上的奇点的重数 舒伯特变奏曲 写下余法线的方程 舒伯特簇的一束 研究变量XY = YX = 0 其中X和Y是矩阵。 发展标准单项理论 对于与不可分解的，可对称化的， 仿射型广义Cartan矩阵 这项研究是在代数几何，研究的 几何对象（变种）作为系统的解决方案而产生， 多项式方程 私家侦探考虑非常具体的品种 并要求非常明确的组合答案。 的 她认为品种的类型是非常重要的， 她所寻求的答案是广泛适用的。 非常重要 这项研究将会有结果。

This research will concentrate on problems connected with a semi-simple, simply-connected algebraic group G over a field k, the Borel subgroup B of G and a Schubert variety in the flag variety G/B. The P.I. has developed a standard monomial theory as a generalization of the classical Hodge-Young standard monomial theory. The following specific problems will be considered. Determine the multiplicity of a singular point on a Schubert variety. Write down the equations of the conormal bundle of a Schubert variety. Study the variety XY = YX = 0 where X and Y are matrices. Develope a standard monomial theory for a Kac group associated to an indecomposable, symmetrizable, generalized Cartan matrix of affine type. This research is in algebraic geometry, the study of the geometric objects (varieties) arising as solutions of systems of polynomial equations. The P.I. considers very concrete varieties and asks for very explicit, combinatorial answers to them. The types of varieties she considers are very important and the type of answers she seeks are widely applicable. Very important results will come out of this research.