Tensor Product Decompositions of Group Representations

Sammanfattning

This thesis studies the representation theory of finite groups, with an emphasis on tensor product decompositions. We begin by introducing the basic concepts of group representations, modules, homomorphisms and equivalence of representations. We then develop the theory of characters, inner products of characters, and the decomposition of the group algebra. The symmetric group S3 is used throughout as a running example to illustrate the theory. We then investigate the irreducible representations of S4. We first compute the character table of S4 followed by finding its Clebsch-Gordan table. By these calculations, we derive a general formula for the number of simple components contained in a k-tensor product of simple modules of S4. These results illustrate key ideas in representation theory and provide a concrete introduction to more general methods.