Representations and Invariants of the Classical Groups (Encyclopedia of Mathematics and its Applications)

Rating:
Summary: typical math text
Review: Sadly, this book does not live up to its
reputation and can only be used by those
already having a working knowledege of
the subject. On top of that, the printing
is atrocious in places - unacceptable for
an expensive item.

Rating:
Summary: typical math text
Review: The book may be considered an update of Weyl's classic from 1946 "The classical groups, their invariants and representations". Naturally the theory has developed immensly since then and a new comprehensive presentation has been urgently overdue. Goodman and Wallach are not only among the most qualified to write this next "classic" on the subject, they also put a phantastic effort into making it comprehensive, readable, and pleasant to use. I found it easy to find needed results and concrete formulae on decompositions of tensors and harmonics, duailty groups and centralizer algebras, mulitiplicity formulae, branching laws, etc.. The results are systematically organized - almost in an "encyclopedial" style. The book also contains rather modern results, such as, for example, new proofs of the Kostant Rallis Theorem or the relations between invariant theory and the Jones knot polynomial.

This new "classic" sets the standard and foundations for any mathematician working in the field. I consider it especially useful also for mathematician from other fields as well as theoretical physicists who want to apply invariant theory in a variety of concrete settings.

Rating:
Summary: Comprehensive, well organized, very applicable
Review: The book may be considered an update of Weyl's classic from 1946 "The classical groups, their invariants and representations". Naturally the theory has developed immensly since then and a new comprehensive presentation has been urgently overdue. Goodman and Wallach are not only among the most qualified to write this next "classic" on the subject, they also put a phantastic effort into making it comprehensive, readable, and pleasant to use. I found it easy to find needed results and concrete formulae on decompositions of tensors and harmonics, duailty groups and centralizer algebras, mulitiplicity formulae, branching laws, etc.. The results are systematically organized - almost in an "encyclopedial" style. The book also contains rather modern results, such as, for example, new proofs of the Kostant Rallis Theorem or the relations between invariant theory and the Jones knot polynomial.

This new "classic" sets the standard and foundations for any mathematician working in the field. I consider it especially useful also for mathematician from other fields as well as theoretical physicists who want to apply invariant theory in a variety of concrete settings.