Representation Theory of Finite Groups with Examples

Karthik, Timmavajjula Venkata (2018) Representation Theory of Finite Groups with Examples. Masters thesis, Indian Institute of Science Education and Research Kolkata.

PDF (MS dissertation of Timmavajjula Venkata Karthik (13MS091)) 13MS091.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

This thesis involves the study of representation theory of finite groups over complex vector space along with examples of the symmetric groups S₃, S₅ and the group GL₂(Fq), where Fq denotes the finite field of characteristic not equal to two. The study is based mainly on the texts Linear Representations of Finite Groups by Jean-Pierre Serre and Introduction to Representation Theory by Pavel Etingof et al. In chapter 1 we look at the basic notions of representation theory of finite groups and study the example of S₃. To understand it deeper, we take a closer look at the irreducible representations of a group and their properties in chapter 2. We prove the non-trivial result that the number of irreducible representations of a finite group up to isomorphism is equal to the number of conjugacy classes of the group. We also see that the theory of irreducible representations is trivial for an abelian group and enumerate the one dimensional representations of a group upto isomorphism. At the end of chapter 2 we study the example of S₅. In chapter 3, we introduce the notions of induced and virtual representations and prove the Frobenius reciprocity theorem. Using the results of chapter 3, we study in detail the representation theory of the group GL₂(Fq) in chapter 4.

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