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Representation Theory of Finite Groups and Associative Algebras

Charles W. Curtis and Irving Reiner
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

I guess I'm officially an old fogey now that a book that was recommended to me in graduate school as a standard reference has been republished by AMS/Chelsea. After all, this is the publisher (now taken over by AMS) who made its name by publishing old classics in (fairly) inexpensive hardcover editions. Curtis and Reiner's famous account of representation theory certainly deserves to be considered a classic, and its return should be greeted with delight by everyone.

As I was working on this review, my group theorist colleague Tom Berger saw it on my desk, and said "That's the book! That's the book I memorized in graduate school." I think a lot of people will react in the same way. For a very long time, "Curtis and Reiner" was where you went to learn this subject. It was well written and contained almost all that was important to know about the basics of representation theory.

It's not a perfect book. The section on modular representation theory at the end is too short, for example, and very classical, very close to Brauer's original work. There are no infinite groups at all, so that those whose interest in representation theory comes from physics or from the Langlands program will have to go elsewhere.

The authors themselves recognized some of these problems, and went on to write a massive second book, Methods of Representation Theory: With Applications to Finite Groups and Orders, which filled two volumes and never quite caught on like their first book. It was last reprinted in the "Wiley Classics Library", but seems now to be out of print.

Nevertheless, while today's students may well want to start with, say, Fulton and Harris's Representation Theory : A First Course, they will still find much useful material in this book. I'm certainly glad to have it back.

Fernando Q. Gouvêa never did become a representation theorist, preferring to move to number theory and history of mathematics. He is now professor of mathematics at Colby College in Waterville, ME.

The table of contents is not available.