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Douglas_Knight comments on The Best Textbooks on Every Subject - Less Wrong

167 Post author: lukeprog 16 January 2011 08:30AM

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Comment author: [deleted] 16 January 2011 05:30:02PM *  19 points [-]

Subject: Representation Theory

Recommendation: Group Theory and Physics by Shlomo Sternberg.

This is a remarkable book pedagogically. It is the most extremely, ridiculously concrete introduction to representation theory I've ever seen. To understand representations of finite groups you literally start with crystal structures. To understand vector bundles you think about vibrating molecules. When it's time to work out the details, you literally work out the details, concretely, by making character tables and so on. It's unique, so far as I've read, among math textbooks on any subject whatsoever, in its shameless willingness to draw pictures, offer physical motivation, and give examples with (gasp) literal numbers.

Math for dummies? Well, actually, it is rigorous, just not as general as it could potentially be. Also, many people's optimal learning style is quite concrete; I believe your first experience with a subject should be example-based, to fix ideas. After all, when you were a kid you played around with numbers long before you defined the integers. There's something to the old Dewey idea of "learning by doing." And I have only seen it tried once in advanced mathematics.

Fulton and Harris won't do this. The representation theory section in Lang's Algebra won't do this -- it starts about three levels of abstraction up and stays there. Weyl's classic The Theory of Groups and Quantum Mechanics isn't actually the best way to learn -- the group theory and the physics are in separate sections and both are a little compressed and archaic in terminology. Sternberg is really a different thing entirely: it's almost more like having a teacher than reading a textbook.

The treatment is really most relevant for physicists, but even if you're not a physicist (and I'm not), if you have general interest in math, and background up to a college abstract algebra course, you should check this out just to see what unusually clear, intuitive mathematical writing looks like. It will make you happy.

Comment author: Douglas_Knight 17 January 2011 09:38:56PM 3 points [-]

Fulton and Harris won't do this.

Won't do what? Almost everything you say about Sternberg seems to me to apply to Fulton & Harris. I have not looked at Sternberg, and it may well be better in all these ways, but your binary dismissal of F&H seems odd to me.