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

对称群的表示理论已经研究了一个多世纪，但许多基本问题仍未解决。其中一个问题是理解对称群的两个简单模的体积，一个关于对称群的表示和两个对称群的环积表示的构造。由此产生的体积系数“可能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）。图解代数是一个较新的主题，这种代数出现在数学和物理的不同部分，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来解决关于对称群及其环积的问题。在这个专题中，学生将使用表示理论和组合方法来研究这些图代数，并推导出理解plethysm的结果。

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.