2 1. Introduction
graduate students and should be accessible to students with a strong
background in linear algebra and a basic knowledge of abstract al-
gebra. Theoretical material in this book is supplemented by many
problems and exercises which touch upon a lot of additional topics;
the more difficult exercises are provided with hints.
The book covers a number of standard topics in representation
theory of groups, associative algebras, Lie algebras, and quivers. For
a more detailed treatment of these topics, we refer the reader to the
textbooks [S], [FH], and [CR]. We mostly follow [FH], with the
exception of the sections discussing quivers, which follow [BGP], and
the sections on homological algebra and finite dimensional algebras,
for which we recommend [W] and [CR], respectively.
The organization of the book is as follows.
Chapter 2 is devoted to the basics of representation theory. Here
we review the basics of abstract algebra (groups, rings, modules,
ideals, tensor products, symmetric and exterior powers, etc.), as well
as give the main definitions of representation theory and discuss the
objects whose representations we will study (associative algebras,
groups, quivers, and Lie algebras).
Chapter 3 introduces the main general results about representa-
tions of associative algebras (the density theorem, the Jordan-H¨older
theorem, the Krull-Schmidt theorem, and the structure theorem for
finite dimensional algebras).
In Chapter 4 we discuss the basic results about representations of
finite groups. Here we prove Maschke’s theorem and the orthogonality
of characters and matrix elements and compute character tables and
tensor product multiplicities for the simplest finite groups. We also
discuss the Frobenius determinant, which was a starting point for
development of the representation theory of finite groups.
We continue to study representations of finite groups in Chapter
5, treating more advanced and special topics, such as the Frobenius-
Schur indicator, the Frobenius divisibility theorem, the Burnside the-
orem, the Frobenius formula for the character of an induced repre-
sentation, representations of the symmetric group and the general
Previous Page Next Page