TITLE: Representation theory and elementary particles
QUESTION [35 upvotes]: I have been looking for a clear  expository mathematical text on the relation between the theory of elementary particles and the representation theory of $U(1), SU(2), SU(3)$, I was very disappointed about The step which explains why an elementary particle should correspond to an irreducible representation!
I asked physicists to explain The step and I get more confused then before asking. I was wondering if someone could explain this passage with mathematical rigor and physical principles! And pretty sure I'm not the only one in this situation :) 
Thanks.
Edit Thank you very much all for your answers, unfortunately I can't choose one! It was great with lot of gratitude to all of you, still I do not understand the key step probably I need to learn more physics to feel what is really happening.

REPLY [4 votes]: The answers given so far are all beautiful, but lack one detail which I think should be clarified. I'll try to answer the question concerning The Step in a more intuitive way, but essentially following John Baez as quoted by Carlo Beenakker. 
First of all, the definition of elementary particle is not entirely rigorous. In high energy physics, the phrase elementary particle is unfortunately used for truly elementary particles (these are stable particles, which do not decay into others) and quite a lot of unstable particles. 
Let me start with the stable elementary particles. The very fact that they are called elementary means that all mathematical structures needed to describe their properties should be such that these cannot be decomposed into smaller entities. Otherwise, it would be extremely likely that the particle described with these mathematical structures could also be decomposed into "smaller" particles, it would not be stable, or it could at least be fragmented by external forces. Thus, if properties of particles are related to representations of (Lie) groups, stable elementary particles should be related to the smallest possible representations, i.e. the fundamental representations (which happen to be irreducible). 
Now, all the other particles, which happen to be not stable, will ultimately decay into stable elementary particles. Thus, they should be related to representations which can be decomposed into fundamental ones. A particle can only decay into other particles, if the tensor product of all the representations related to the fragments contains the representation of the original particle. 
High energy experiments produce quite a lot of states which physicists like to associate with particles or excited states of particles. Since the compact Lie groups, which typically appear in physics, have the property that all of their finite dimensional representations can be decomposed into direct sums of irreducible representations, it became customary to associate the word particle in the sense of building block of physical entities to irreducible representations. But as explained above, only stable elementary particles are naturally associated with particular representations, the fundamental ones. For the others, it is merely a convenient way to get some order into the zoo of high energy physics. As Baez writes, working with the irreps is a convenient choice of basis.
Finally, physicists nowadays use the word quasiparticle for a lot of physical entities, which behave like particles, i.e. like excitations of quantum fields of some effective quantum field theories, in particular in condensed matter systems. Again, the word is defined to refer to entities which have well defined representation theoretic properties with respect to all groups involved. 
I know, this is a very sloppy answer. However, I found it important to shift the focus towards the important distinction between truly elementary stable particles and all the other "elementary" particles.