# Importance of Representation Theory

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 must certainly be 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 $$SO(3)\mathrm{SO}(3)$$ 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 $$GL(n)\mathrm{GL}(n)$$ and then extend it to act on wave functions by $$ψ(x,t)↦ψ(Ax,t)\psi(x,t) \mapsto \psi(Ax,t)$$ for $$AA$$ in $$SO(n)\mathrm{SO}(n)$$.

This Wikipedia article on particle physics and representation theory claims that if our physical system has $$GG$$ as a symmetry group, then there is a correspondence between particles and representations of $$GG$$. 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 $$GG$$ action? Anyone know of good sources that talk about this?

## Answer

One comment about your sentence “this seems to only trivially invoke representation theory”. It might be surprising, but such obvious representations are actually the source of interesting mathematics, and a lot of effort of representation theorists is devoted to studying them.

More precisely: start with a group (in your example $$SO(n)SO(n)$$) acting on a space $$XX$$ (in your example $$Rn\mathbb R^n$$), and look at the space of functions on $$XX$$ (let me write it $$F(X)\mathcal F(X)$$; in a careful treatment, one would have to think about whether we wanted continuous, smooth, $$L2L^2$$, or some other kind of functions, but I will suppress that kind of technical consideration).

Then, as you observe, there is a natural representation of $$GG$$ on $$F(X)\mathcal F(X)$$.

You are right that from a certain point of view this seems trivial, because the representation is obvious. Unlike when one first learns rep’n theory of finite groups, where one devotes a lot of effort to constructing reps., in this context, the rep. stares you in the face.

So how can this be interesting?

Well, the representation $$F(X)\mathcal F(X)$$ will almost never be irreducible. How does this representation decompose?

Suddenly we are looking at a hard representation theoretic problem.

• First, we have to work out the list of irreps of $$GG$$ (which is much
like what one does in a first course on rep’n theory of finite groups).

• Second, we have to figure out how $$F(X)\mathcal F(X)$$ decomposes, which involves representation theory (among other things, you have to develop methods for investigating this sort of question), and also often a lot of analysis (because typically $$F(X)\mathcal F(X)$$ will be infinite dimensional, and may be a Hilbert space, or have some other similar sort of topological vector space structure which should be incorporated into the picture).

I don’t think I should say too much more here, but I will just give some illustrative examples:

1. If $$X=G=S1 X = G = S^1$$ (the circle group, say thought of as $$R/Z\mathbb R/\mathbb Z$$)
acting on itself by addition, then the solution to the problem of decomposing
$$F(S1)\mathcal F(S^1)$$ is the theory of Fourier series. (Note that a function on $$S1S^1$$ is the
same as a periodic function on $$R\mathbb R$$.)

2. If $$X=G=R X = G = \mathbb R$$, with $$GG$$ acting on itself by addition, then
the solution to the above question (how does $$F(R)\mathcal F(\mathbb R)$$ decompose under the
action of $$R\mathbb R$$) is the theory of the Fourier transform.

3. If $$X=S2 X = S^2$$ and $$G=SO(3)G = SO(3)$$ acting on $$XX$$ via rotations, then decomposing $$F(S2)\mathcal F(S^2)$$ into irreducible representations gives the theory of spherical harmonics.
(This is an important example in quantum mechanics; it comes up for example in the theory of the hydrogen atom, when one has a spherical symmetry because the electron orbits the nucleus, which one thinks of as the centre of the sphere.)

4. If $$X=SL2(R)/SL2(Z) X = SL_2(\mathbb R)/SL_2(\mathbb Z)$$ (this is the quotient of a Lie group by a discrete subgroup, so is naturally a manifold, in this case of dimension 3),
with $$G=SL2(R)G = SL_2(\mathbb R)$$ acting by left multiplication, then the problem of decomposing $$F(X)\mathcal F(X)$$ leads to the theory of modular forms and Maass forms, and is the first example in the more general theory of automorphic forms.

Added: Looking over the other answers, I see that this is an elaboration on AD.’s answer.

Attribution
Source : Link , Question Author : Eric O. Korman , Answer Author : Mike Pierce