Oy! YABOGR—yet another book on group representations. The theory is so
beautiful and so central to mathematics that it tends to draw authors like honey
draws flies. Given the existence of several excellent monographs (among which I'd
include Adams [1], Fulton-Harris [5], Samelson [17], and Serre [19]), why do I feel
this book a worthy addition to the textbook literature on the subject?
I think two facets distinguish my approach. First, this book is relatively ele-
mentary; and second, while the bulk of books on the subject (and, in particular,
the four quoted above) are written from the point of view of an algebraist or a
geometer, this book is written with an analytical flavor.
As for the elementary nature of the material, much of it is self-contained. A
prior exposure to the notions of quotient group and the isomorphism theorems is
assumed; but, for example, I develop the necessary theory of algebraic integers in
proving that the dimension of an irreducible representation divides the vector of
the group. The material on finite groups (Chapters I-VI) should be suitable for an
upper-level undergraduate course, either as a separate course or as a supplement
to an advanced algebra course.
The material on compact groups is a little more sophisticated but I have discussed
the needed calculus on manifolds and have even included an appendix on the basic
theory of self-adjoint Hilbert-Schmidt operators, which is needed to prove the Peter-
Weyl theory.
However, there are a few places we need some facts from algebraic topology that
we used without proof: for example, the exact sequence of a fibration.
As for the analyst's point of view, much of the most profound work on group
representations has been done by analysts: Weyl, Gel'fand, and Mackey come to
mind. Indeed, this monograph bears a strong influence of George Mackey from
whom I first learned much of the material thirty years ago.
The analyst's approach can be seen in several places: for example, the last three
chapters discuss the structure and representations of the compact groups, not the
representations of the semisimple Lie algebras. The two are closely related but
the former is more elementary and decidedly less algebraic. In this regard, the
discussion is closest to that of Adams [1].
A critical role is played by the fact that for compact groups, it is easy to show that
any (finite-dimensional, continuous) representation supports an invariant positive
definite inner product. This immediately implies that on the Lie algebra, the adjoint
representation is by matrices which are skew-adjoint in a suitable inner product,
so the operators are semisimple and the Killing form is (strictly) negative definite.
This replaces pages of algebraic minutiae.
Previous Page Next Page