Varieties Related to Algebraic Group Actions

摘要：韦曼这份提案由几个部分组成。它们都与与代数群和表示理论有关的簇的研究有关。第一部分是研究与代数群有关的簇所支持的等变模，如行列式簇和单李代数中幂零共轭类的闭包。作者建议研究这类模范畴的Grothendieck群，并利用这些结果通过退化序列来研究轨道闭合的坐标环。第二部分是广义判别式和结式的奇异性研究。这位研究者建议将他对超定数簇的不可约分支的分类(与Zlevinsky一起)扩展到可约群的任意齐次空间。第三部分是关于Z_2分次李代数gl(m，n)的表示。作者建议研究这种表示与一般$m×n$矩阵的多项式环中等变理想合子之间的关系。这项建议的主题是研究与经典群有关的变种。一般说来，它们可以被描述为具有许多对称性的品种。这些变体在交换代数中扮演着重要的角色，既是基本的例子，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被证明给出了这类变元。研究者建议研究与这些簇相关的代数不变量，特别是它们所支持的等变模的结构。

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.