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Group Representation for Physicists

Jin-Quan Chen, Jialun Ping, and Fan Wang
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
, on

The fact that Group Representation Theory for Physicists is now in its second edition is a signal that there is a real audience for books like this. And that is interesting. It reflects the fact that group representation theory continues to be an important part of theoretical physics. It also reflects the fact that physicists find it very difficult to read the books we mathematicians write.

This book is an introduction to groups and representation theory aimed at physicists. It starts with the definition of a group, and already on that page there are things that I find strange. A group is a set of elements (or operators) with an operation that satisfies four conditions: closure, associativity, existence of an identity element, existence of inverses. Fine. Then: "Since in general ab is not equal to ba the order of multiplication is important. An Abelian group is one whose elements commute with one another, that is [a, b] = ab - ba = 0." So right off the bat there seems to be present another operation, called "-". In other words, these groups really are subgroups of an algebra of operators from the beginning.

And so it goes: the notation is different, the examples are different, some of the words are different. There are lots of formulas, and most symbols seem to have at least two indices on them. (I think I saw one with four subscripts and two superscripts.) Despite its strangeness to someone who comes from "another tradition," in the end this seems all to the good. These physicists have made this mathematics their own and developed their own way of understanding it. I suspect that we can learn something by reading their take on it all.


Fernando Q. Gouvêa ( is the author of several books, including, most recently, Math through the Ages, written in collaboration with William Berlinghoff.

  • Elements of Group Theory
  • Group Representation Theory
  • Representation Theory for Finite Groups
  • Representation Theory of the Permutation Group
  • Lie Groups
  • The Rotation Group
  • The Unitary Groups
  • The Point Groups
  • Applications of Group Theory to Many-Body Systems
  • The Space Groups