Representation Theory of Finite Groups: An Introductory Approach

Description

This book is intended to present group representation theory at a level accessible to mature undergraduate students and beginning graduate students. This is achieved by mainly keeping the required background to the level of undergraduate linear algebra, group theory and very basic ring theory. Module theory and Wedderburn theory, as well as tensor products, are deliberately avoided. Instead, we take an approach based on discrete Fourier Analysis. Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject. A number of exercises are included. This book is intended for a 3rd/4th undergraduate course or an introductory graduate course on group representation theory. However, it can also be used as a reference for workers in all areas of mathematics and statistics. Front Matter....Pages i-xiii Introduction....Pages 1-1 Review of Linear Algebra....Pages 3-11 Group Representations....Pages 13-25 Character Theory and the Orthogonality Relations....Pages 27-50 Fourier Analysis on Finite Groups....Pages 51-70 Burnside’s Theorem....Pages 71-82 Group Actions and Permutation Representations....Pages 83-96 Induced Representations....Pages 97-109 Another Theorem of Burnside....Pages 111-115 Representation Theory of the Symmetric Group....Pages 117-129 Probability and Random Walks on Groups....Pages 131-152 Back Matter....Pages 153-157