Mathematical Sciences: Schubert Varieties and Standard Monomial Theory

本文主要研究域k上的半单、单连通代数群G、G的Borel子群B以及旗簇G/B中的Schubert簇的问题。P.I.发展了一个标准单项理论作为经典的Hodge-Young标准单项理论的推广。将考虑以下具体问题。确定Schubert簇上奇点的重数。写下舒伯特簇的余法丛的方程。研究簇XY=YX=0，其中X和Y是矩阵。给出了仿射型不可分解、对称化、广义Cartan矩阵的Kac群的标准单项理论。这项研究是在代数几何中进行的，研究作为多项式方程组的解而出现的几何对象(簇)。P.I.考虑了非常具体的变化，并要求对它们进行非常明确的组合回答。她考虑的品种类型非常重要，她寻求的答案类型广泛适用。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.