Representations of symmetric groups, wreath products of symmetric groups and related diagram algebras

对称群的表示、对称群的花圈积及相关图代数

• 批准号: 2289820
• 金额: --
• 依托单位: University of Kent
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2289820
• 关键词: Representations; symmetric; groups; wreath; products

The representation theory of symmetric groups has been studied for over a century but many fundamental questions remain open. One such problem is to understand the plethysm of two simple modules of symmetric groups, a construction relating representations of symmetric groups and representations of wreath products of two symmetric groups. The resulting plethysm coefficients are "perhaps the most challenging, deep and mysterious objects in algebraic combinatorics" (Pak, Panova 2017). Diagram algebras are a more recent subject with such algebras arising in different parts of mathematics and physics, brought together by Graham and Lehrer's definition of cellular algebras in 1995. But certain diagram algebras provide an important tool with which to attack questions about symmetric groups and their wreath products. In this project the student will use representation-theoretic and combinatorial methods to investigate these diagram algebras and derive consequences for understanding plethysm.

对称群的表示论已经被研究了一个多世纪，但许多基本问题仍然悬而未决。其中一个这样的问题是理解对称群的两个简单模块的体积，一种与对称群的表示和两个对称群的花环乘积的表示相关的构造。由此产生的体积系数“也许是代数组合学中最具挑战性、最深刻和最神秘的对象”（Pak，Panova 2017）。图代数是一门较新的学科，此类代数出现在数学和物理学的不同部分，由 Graham 和 Lehrer 在 1995 年对细胞代数的定义汇集在一起​​。但是某些图代数提供了一个重要的工具，可以用来解决有关对称群及其对称群的问题。花环产品。在这个项目中，学生将使用表示论和组合方法来研究这些图代数并得出理解体积的结果。