Symmetry in terms of matrices When we encounter a problem in physics which can be expressed in terms of matrices or tensors, why do we decompose the tensor in terms of its symmetric and antisymmetric or trace components? What is the physics motivation behind doing so?
 A: Usually this is done because the various pieces in the decomposition transform "nicely".  For instance, the pieces that are symmetric will usually only transform amongst themselves, and likewise for the antisymmetric pieces.  This decomposition thus often facilitates bookkeeping in calculations.
The process is similar in spirit to decomposing many-particle spin 1/2 states into states of definite total net spin: in this case states of a given net total spin $S$ transform amongst themselves.
A practical byproduct of such decompositions is that some terms may cancel out "by symmetry", just like - say - some integrals are obviously $0$ since the integrand is odd but the integration interval is symmetric.
As another application, selection rules might also allow or disallow some processes to occur based only on symmetry arguments, in which case only the properly symmetrized pieces remain.
A: OP is basically asking: 

Why do we decompose (reducible) group representations in irreducible group representations? 

Partial answer:


*

*To classify the (reducible) representation. 

*Irreducible representations can not be further truncated without destroying the group symmetry.

*Because certain irreducible sub-representations of the given (reducible) representation may be forbidden by e.g. selection rules, other physical principles, etc, and this is always useful information.
