Customer Reviews

The Classical Groups: Their Invariants and Representations

5 star:		(2)
4 star:		(0)
3 star:		(1)
2 star:		(0)
1 star:		(0)

 

 

Forget about that pansy abstract axiom approach. This is the WWF of group theory. Weyl will take anyone to the mat with this book. It is packed with detail and demonstrations. He follows the vector space/matrix representation approach common to digital systems, physics and chemistry rather than axiomatic, generators/permutations approach more common in Abstract Algebra courses. This is the lineage that develops matrix transforms as groups starting from "the full group of all non-singular linear transformations and .. the orthogonal groups" (p. vii). The latter chapters cover characters and invariants. Galois and field theory have been vanquished. Chapter 2, "Remembrance of things past" is very entertaining. My favorite quote, "Here there is only one man to mention - Hilbert. His papers (1890/92) mark a turning point in the history of invariants theory. He solves the main problems and thus almost kills the whole subject." It's funny because it's true. This is almost a botanical treatise in which the matrix groups are studied as specimens in the jungle -- "...after all each group stands in its own right and does not deserve to be looked upon merely as a subgroup of...Her All-embracing Majesty GL(n)." (p 136). Historic references throughout provide motivation and entertainment. You couldn't possibly be disappointed with this book.

Although this is a dated work, lacking some of the more modern language, it is still worth owning and reading. It is, after all, a designated "classic." And the material presented has been incorporated within so many aspects of physics that one simply cannot avoid needing a book such as this. There are better books on the subject, for both mathematicians and physicists, but this book still proves its worth.

His presentation of simplectic groups ( although short) was helpful.
That said having had Weyl's Space, Time, Matter for many years
I'm used to his notation, but here he is very intense in his presentation
and somewhat less than clear. As he is the founder of gauge group theory
one expects some mathematics, just not where it seems more difficult than necessary?
Also the, now, standard classifications of Cartan groups aren't mentioned ( although might be
because of the original publishing date of 1939?).
I got this book looking for the Weyl root groups used in making Cartan invariant matrices
which I couldn't find.
So on these points the book is a disappointment while still being a classic text.