Varieties Related to Algebraic Group Actions

摘要，韦曼 这项建议包括几个部分。它们都与 与代数群和表示论有关的对簇的研究。 第一部分是研究单李代数中与代数群有关的簇所支持的等变模，如行列式簇和幂零共轭类的闭包。 研究者建议调查的Grothendieck群的类别，这样的模块，并应用这些结果来研究坐标环的轨道闭包的退化序列。 第二部分是对《红楼梦》的研究。 广义判别式和广义结式的奇异性。 调查人员建议延长他的分类（与Zelevinsky）的不可约成分的hyperdeterminantal品种的任意齐性空间的还原群。 第三部分研究Z_2分次李代数gl（m，n）的表示。 调查员建议研究这种表示与syzygies的等变理想的多项式环的通用m\倍n$矩阵的连接。 这个提议的主题是研究与经典群相关的变体。 一般来说，它们可以被描述为具有许多对称性的变种。 这些变种在交换代数中起着重要的作用，作为基本的例子，也提供了与其他领域的联系，如组合学，表示论和奇点。历史上许多重要的一般定理首先证明了这些种类的品种。 研究人员建议研究与这些品种相关的代数不变量，特别是结构 支持的同变模块。

ABSTRACT, WEYMAN This proposal consists of several parts. They are all connected to the study of varieties related to algebraic groups and representation theory. The first part is to study equivariant modules supported in varieties related to algebraic groups, like determinantal varieties and the closures of nilpotent conjugacy classes in simple Lie algebras. The investigator proposes to investigate the Grothendieck groups of categories of such modules and to apply these results to study the coordinate rings of orbit closures by means of degeneration sequences. The second part is the study of singularities of generalized discriminants and resultants. The investigator proposes to extend his classification ( with Zelevinsky ) of irreducible components of hyperdeterminantal varieties to arbitrary homogeneous spaces for the reductive group. The third part is concerned with representations of Z_2 graded Lie algebras gl(m,n). The investigator proposes to study the connection of such representations with syzygies of equivariant ideals in the polynomial ring of generic $m\times n$ matrices. The subject of this proposal is the study of the varieties related to classical groups. In general terms they can be described as the varieties with a lot of symmetries. These varieties play an important role in Commutative Algebra, serving as basic examples and also providing links to other fields, like Combinatorics, Representation Theory and Singularities. Historically many important general theorems were proved first for these kinds of varieties. The investigator proposes to study the algebraic invariants associated to these varieties, in particular the structure of equivariant modules supported in them.