Mathematical Sciences: Schubert Varieties and Standard Monomial Theory

本研究将集中研究域k上的半简单、单连通代数群G、G的Borel子群B和标志群G/B中的Schubert变种所涉及的问题。作为经典霍奇-杨标准单项式理论的推广，P.I.发展了一个标准单项式理论。将考虑以下具体问题。确定舒伯特变上一个奇点的多重性。写出舒伯特变换的法向束的方程。研究变量XY = YX = 0，其中X和Y是矩阵。发展与不可分解、可对称的仿射型广义Cartan矩阵相关的Kac群的标准单项式理论。这门学科属于代数几何，研究作为多项式方程组解的几何对象（变量）。P.I.考虑非常具体的变量，并要求非常明确、组合的答案。她考虑的品种类型是非常重要的，她寻求的答案类型是广泛适用的。这项研究将得出非常重要的结果。

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.