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Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I can think of two mathematical reasons for studying it:

1) The character table of a group is packs a lot of information about the group and is concise.

2) It is practically/computationally nice to have explicit matrices that model a group.

But there are for sure deeper things that I am missing. I can understand why one would want to study group actions (the axioms for a group beg you to think of elements as operators), but why look at group actions on vector spaces? Is it because linear algebra is so easy/well-known (when compared to just modules, say)?

I am also told that representation theory is important in quantum mechanics. For example, physics should be SO(3) invariant and when we represent this on a Hilbert space of wave-functions, we are led to information about angular momentum. But this seems to only trivially invoke representation theory since we already start with a subgroup of GL(n) and then extend it to act on wave functions by $\psi(x,t) \mapsto \psi(Ax,t)$ for A in SO(n).

This http://en.wikipedia.org/wiki/Particle_physics_and_representation_theory wikipedia article claims that if our physical system has G as a symmetry group, then there is a correspondence between particles and representations of G. I'm not sure if I understand this correspondence since it seems to be saying that if we act an element of G on a state that corresponds to some particle, then this new state also corresponds to the same particle. So a particle is an orbit of the G action? Anyone know of good sources that talk about this?

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    What do you mean by SO(3)?2010-07-24
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    Linear algebra is often much nicer - that is just what it comes down to2010-07-24
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    @Casebash informally its another way of saying "all the ways you can rotate something" http://en.wikipedia.org/wiki/Rotation_group2010-07-24
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    great question! It took me about 16 times of seeing the basics of representations before I had any idea what they were good for.2010-07-24
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    Much of math is reducing hard things to linear algebra. For examples: calculus (differentiating is locally replacing your function by the best linear function available), representation theory, homology.2010-07-24
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    http://uregina.ca/~mareal/flag-coh.pdf http://yufeizhao.com/papers/youngtab-hcmr.pdf http://www.math.ku.edu/~jmartin/courses/math824-F10/ https://projecteuclid.org/euclid.lnms/1215467407 just a couple pointers to representation theory that may be interesting. Not physical though.2017-04-07

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