Explaining a quote by Weinberg about the significance of symmetry groups in physics When skimming through a book, I found this quote:

The universe is an enormous direct product of representations of symmetry
  groups. —Steven Weinberg

I am a mathematician (so I know only basic high-school-level physics) but I know a little about symmetry groups and about representations of groups. I am wondering if someone could explain what motivated the above quote. To me representations seem like very abstract objects which should be completely unrelated to the real world, so I'm naturally very interested to find out how this is connected to physics. 
 A: You probably know that there are some fundamental symmetries to physical laws. But to define the action of the symmetry, there must be an object to apply it to. Such an object is most naturally mathematically expressed as a representation of the respective symmetry group.
Weinberg has two distinct cases on his mind, particles (particle fields) as Lorentz-group representations and gauge fields (i.e. conservation laws/symmetries of particle interaction) as internal symmetry representations. From these two ingredients, the Standard model of particle physics is built. I.e. you can understand the current model of all interactions of matter and energy (up to gravity) as a theory whose configuration space is a direct product of symmetry representations.
(Gravity can be also understood as the (1,1) representation of the Lorentz group but the direct-productness with the Standard model and the whole gauge-field nature of the theory is disputable.)

As to the whole universe, the standard cosmology models the spatial slice of the universe as a maximally symmetric manifold, i.e. the trivial representation of the group of translations and rotations. Here, however, you have to take Weinberg with a grain of salt since the maximal symmetry of the spatial slice is only applicable from $\sim Mpc$ scales up and there is no fundamental argument why it should be a representation of the group of rotations and translations.
A: I give my shot:
A modern point of view of the mathematics of quantum theories is that the observables (objects you can measure) of the theory are self-adjoint operators affiliated to a von Neumann algebra $V$. An operator is affiliated to a von Neumann algebra if all its spectral projections are in the algebra (or, if bounded, if it is itself in algebra).
In order for $V$ to be an algebra relevant to describe a physical system, it should contain the suitable representation of certain symmetry groups: a basic example would be the (possibly infinite dimensional) Heisenberg group associated to the canonical commutation relations.
Now a physical system is then implemented by a state $\omega$ of the algebra $V$ (a positive element of its dual $V^*$ with norm one); and $\omega(v)$, $v\in V$, physically represents the average value of $v$ on the state $\omega$ (or if $v$ is a spectral projection corresponding to an observable $w$, it gives the probability of measuring a determined range of values for $w$ in the state $\omega$). To this state $\omega$ it is associated, via the GNS construction, a cyclic representation $(\pi_\omega,H_\omega, \Omega_\omega)$ of the algebra $V$ as bounded operators $\pi_\omega(v)$ on the Hilbert space $H_\omega$ (and $\Omega_\omega$ is the cyclic vector). This representation is also a representation of the aforementioned symmetry groups represented in $V$.
Now to construct the "universe" $(\pi, H)$, i.e. the Hilbert space that realizes all possible physical states, one simply takes the direct sum of all these cyclic GNS representations:
$$(\pi,H)=\bigl(\oplus_\omega \pi_\omega \, ,\, \oplus_\omega H_\omega\bigr) \; .$$
This gives the universe of all possible physical states, and by Gelfand-Naimark theorem it is isometrically $*$-isomorphic to the algebra $V$.
This is also, as described above, the direct sum of representations of the symmetry groups. I think he focuses on this aspect because these symmetry groups are really crucial to characterize and describe physical systems. Many times one constructs the algebra $V$ of observables of the theory precisely as the smallest von Neumann algebra that contains the relevant symmetry groups.
A: I think people are perhaps over-thinking this, and that the explanation here is simpler than the other answers may suggest.
Though to a mathematician, representation theory is an obviously rather abstract matter, it's fairly common in mathematical physics to refer to particles as ‘representations’ of their associated symmetry groups.  Thus, for example, the $u$, $d$ and $s$ quarks are fairly commonly referred to as an approximate representation of SU(3).
Thus Weinberg's (or whoever's) remark here is, I think, no more than a slightly whimsical inversion of the idea that the maths is there to explain the physics, by characterising the universe as being there to represent the maths.
