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

本文提出的研究项目是对n × n复矩阵仿射空间的坐标环上有限生成的梯度模的范畴进行组合和代数研究，这些模被支持于幂零共轭类的闭包中，并提供一般线性群的相容作用。这一类的Grothendieck群是由一类特殊的模所产生的，这些模的同型分量具有具有奇妙组合性质的庞加莱多项式。这门学科提供了经典学科之间相互作用的一个美丽的例子，这些经典学科包括共轭类的几何、无特征不变量理论、杨氏表的组合学、量子群的现代表示理论和统计力学中的二维可解晶格模型。本研究属于代数组合学领域，这是数学中的一门学科，它使用离散结构和算法来分析代数系统。由于符号计算通常可以用来帮助解决它的问题，它最近得到了快速的发展和前所未有的成功。这个特别的提议与物理学有关，更具体地说，与统计力学有关。

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.