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In Grand Unified Theories (though I'm sure this a general group theory result) people write the irreducible representations of a group (i.e., the gauge bosons) using a sum of irreducible representations of its subgroup (i.e., the unbroken groups after spontaneous symmetry breaking). For example for the $ {\bf 24} $ of $ SU(5) $ we write, \begin{equation} {\bf 24} = \left( {\mathbf{8}} , {\mathbf{1}} , {\mathbf{0}} \right) + \left( {\mathbf{1}} , {\mathbf{3}} , {\mathbf{0}} \right) + \left( {\mathbf{1}} , {\mathbf{1}} , {\mathbf{0}} \right) + \left( {\mathbf{3}} , {\mathbf{2}} , {\bf \frac{ 5 }{ 3}} \right) + \left( \bar{ {\mathbf{3}} } , {\mathbf{2}} , - {\bf \frac{ 5 }{ 3}} \right) \end{equation} where the notation here is $ \left( SU(3) , SU(2) , U(1) \right) $. Now I understand how to get the first 3 parts since they just arise from the fact that we have a combination of subgroups, but how do you derive the final two? Is this a trivial result or is there some technique (e.g. through Dynkin diagrams) that can be used to extract this for any representation?

I'm ideally looking for a practical technique to perform this breakdown and less for a formal derivation of why it works (unless the derivation is quick, which in my experience in group theory it rarely is).

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Well, Jeff, I assume by now you are out of the woods, but for the odd followup reader now, lets summarize the drill... let's ignore the hypercharge eigenvalue assignments at first (the U(1) eigenvalues, not dimensionalities, in the 3rd entry of this "physics" characterization, not really mathematical, as it does not affect dimensionalities or blocks).

You are breaking up the adjoint of SU(5), so the 24 hermitean traceless 5x5 matrices to blockss transforming uniformly under the 3x3 and 2x2 subspaces. The adjoint of the 3x3 blocks, singlets under the 2x2 transformations is the first (8, 1), the gluons. Symmetrically, the adjoint of the lower-right 2x2 block is the (1,3) adjoint of SU(2) Ts. That leaves 13 independent entries unspanned yet.

One is the obvious traceless diagonal matrix diag(2,2,2,-3-3), un-normalized, acting like the identity in the respective 3- and 2-dim subspaces, so then the (1,1) singlet. The remaining 6+6 pieces hermitean conjugate to each other, are the off-diagonal 3x2 and 2x3 blocks, respectively, so then (3,2) and ($\overline{{\bf3}}$,2).

The hyper-charge Y values are dictated by anomaly and physics considerations, as long as the values for the last two, conjugate blocks, are the opposite of each other. See the related question

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