Some modules of symmetric groups

• Main Author: Jiang, Yu
• Other Authors: Lim Kay Jin
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2021
• Subjects: Science::Mathematics
• Online Access: https://hdl.handle.net/10356/148775
• Institution: Nanyang Technological University

In this thesis, we focus on the representation theory of symmetric groups. Especially, we are very interested in some classical modules of a given symmetric group. Let F be an algebraically closed field of positive characteristic p and Sn be the symmetric group acting on n letters. We first compute the complexities of some simple FSn-modules labelled by two-part partitions. We then consider the classification of trivial source FSn-Specht modules. For this project, the trivial source Specht modules labelled by hook partitions are completely classified. If p > 2, the trivial source Specht modules labelled by two-part partitions are also classified. Moreover, if p = 2, a result for the classification of trivial source Specht modules labelled by partitions with 2-weight 2 is obtained. As an application of this result, a conjecture of [33] is justified. Finally, we generalize a result of Benson and Lim, which motivates us to study the symmetric and exterior powers of Young permutation modules. Along the way, we obtain many results for symmetric and exterior powers of FSn-Young permutation modules. Throughout this thesis, most of the presented results are from [40], [41], [42].