The Combinatorics of Modules Supported in the Closure of a Nilpotent Conjugacy Class

幂零共轭类闭包中支持的模的组合

• 批准号: 9800941
• 负责人: Mark Shimozono
• 金额: $ 4.15万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800941&HistoricalAwards=false
• 关键词: Combinatorics; Modules; Supported; Closure; Nilpotent

9800941 Shimozono The proposed research project is a combinatorial and algebraic study of the category of finitely generated graded modules over the coordinate ring of the affine space of n by n complex matrices, which are supported in the closure of a nilpotent conjugacy class and afford a compatible action of the general linear group. The Grothendieck group of this category is generated by a distinguished family of modules, whose isotypic components have Poincare polynomials with marvelous combinatorial properties. This subject provides a beautiful example of the interplay among classical subjects such as the geometry of conjugacy classes, characteristic-free invariant theory, and the combinatorics of Young tableaux, and the modern representation theory of quantum groups and two-dimensional solvable lattice models in statistical mechanics. This research is in the area of Algebraic Combinatorics, which is a discipline within mathematics which uses discrete structures and algorithms to analyze algebraic systems. It has recently enjoyed rapid growth and unprecedented success, since symbolic computation can often be employed to help solve its problems. This particular proposal has connections with physics, more specifically, with statistical mechanics.

9800941 Shimozono建议的研究项目是关于n×n复矩阵的仿射空间的坐标环上的有限生成分次模范畴的组合和代数研究，它们被支撑在一个幂零共轭类的闭包中，并且提供了一般线性群的相容作用。这一范畴的Grothendieck群是由一族特殊的模生成的，它的同型分支具有具有奇异组合性质的Poincare多项式。这门学科提供了一个很好的例子，说明了经典学科之间的相互作用，如共轭类的几何，无特征不变量理论，Young表的组合学，以及统计力学中量子群的现代表示理论和二维可解晶格模型。这项研究是在代数组合学领域进行的，这是数学中的一门学科，使用离散结构和算法来分析代数系统。它最近获得了快速增长和前所未有的成功，因为符号计算经常可以用来帮助解决它的问题。这一特定的提议与物理学有关，更具体地说，与统计力学有关。