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

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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.