Representation Theory of Finite Groups: An Introductory Approach - Paperback

by Benjamin Steinberg (Author)

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.

Back Jacket

Representation Theory of Finite Groups presents group representation theory at a level accessible to advanced undergraduate students and beginning graduate students. The required background is maintained to the level of linear algebra, group theory, and very basic ring theory and avoids prerequisites in analysis and topology by dealing exclusively with finite groups. Module theory and Wedderburn theory, as well as tensor products, are deliberately omitted. Instead, an approach based on discrete Fourier Analysis is taken, thereby demanding less background from the reader.

The main topics covered in this text include character theory, the group algebra and Fourier analysis, Burnside's pq-theorem and the dimension theorem, permutation representations, induced representations and Mackey's theorem, and the representation theory of the symmetric group. For those students who have an elementary knowledge of probability and statistics, a chapter on random walks onfinite groups serves as an illustration to link finite stochastics and representation theory. Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject and the author provides motivation and a gentle style throughout the text. A number of exercises add greater dimension to the understanding of the subject and some aspects of a combinatorial nature are clearly shown in diagrams.

This text will engage a broad readership due to the significance of representation theory in diverse branches of mathematics, engineering, and physics, to name a few. Its primary intended use is as a one semester textbook for a third or fourth year undergraduate course or an introductory graduate course on group representation theory. The content can also be of use as a reference to researchers in all areas of mathematics, statistics, and several mathematical sciences.

Author Biography

Benjamin Steinberg is a professor at City College of New York and the CUNY graduate center. He received my Ph.D. at the University of California, Berkeley. Steinberg is an algebraist interested in a broad range of areas including semigroups, geometric group theory and representation theory. Other research interests include automata theory, finite state Markov chains and algebras associated to etale groupoids. Steinberg is the co-author of a 2009 Springer publication in the SMM series entitled "The q-theory of Finite Semigroups". This book has had good pre- and post-publication reviews. Ben Steinberg is also an active editorial board member of the Semigroup Forum journal.