Mathematical Sciences: Schubert Varieties and Standard Monomial Theory

数学科学：舒伯特簇和标准单项式理论

• 批准号: 8701043
• 负责人: Venkatramani Lakshmibai
• 金额: $ 3.38万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8701043&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Schubert; Varieties; Standard

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.

这项研究将集中于与域 k 上的半简单、单连通代数群 G、G 的 Borel 子群 B 以及标志变量 G/B 中的舒伯特变量相关的问题。 P.I.发展了标准单项式理论作为经典 Hodge-Young 标准单项式理论的推广。 将考虑以下具体问题。 确定舒伯特簇上奇点的重数。 写出舒伯特簇的共正规丛方程。 研究变量 XY = YX = 0，其中 X 和 Y 是矩阵。 为与不可分解、可对称、广义仿射型嘉当矩阵相关的 Kac 群开发标准单项式理论。 这项研究属于代数几何，研究作为多项式方程组的解而出现的几何对象（种类）。 P.I.考虑非常具体的品种，并要求对它们给出非常明确的组合答案。 她认为品种类型非常重要，她寻求的答案类型也广泛适用。 这项研究将会产生非常重要的结果。