This book is based on courses taught to graduate students at The Institute of Mathematical Sciences, Chennai, and undergraduates of Chennai Mathematical Institute. It presents important combinatorial ideas that underpin contemporary research in representation theory in their simplest setting: the representation theory of symmetric groups, the theory of symmetric functions and the polynomial representation theory of general linear groups. Readers who have a knowledge of algebra at the level of Artin's book [1] (undergraduate honours level) should find this book quite easy to read. However, Artin's book is not a strict pre-requisite for reading this book. A good understanding of linear algebra and the definitions of groups, rings and modules will suffice.

A Chapterwise Description

The first chapter is a quick introduction to the basic ideas of representation theory leading up to Schur's theory of characters. This theory is developed using an explicit Wedderburn decomposition of the group algebra. The irreducible characters emerge naturally from this decomposition. Readers should try and get through this chapter as quickly as possible; they can always return to it later when needed. Things get more interesting from Chapter 2 onwards.

Chapter 2 focusses on representations that come from group actions on sets. By constructing enough such representations and studying intertwiners between them, the irreducible representations of the first few symmetric groups are classified. A combinatorial criterion for this method to work in general is also deduced.

The combinatorial criterion of Chapter 2 is proved using the Robinson– Schensted–Knuth correspondence in Chapter 3. This correspondence is constructed by generalizing Viennot's light-and-shadows construction of the Robinson–Schensted algorithm. The classification of irreducible representations of Sn by partitions of n along with a proof of Young's rule are the main results of this chapter.

Chapter 4 introduces the sign character of a symmetric group and shows that twisting by the sign character takes the irreducible representation corresponding to a partition to the representation corresponding to its conjugate partition.