Group representation of Standard Model On page 527 of Srednicki's textbook "Quantum Field Theory", the Standard Model is described as follows:

It can be succinctly specified as a gauge theory with gauge group $SU(3) \times SU(2) \times U(1)$, with left-handed Weyl fields in three copies of the representation $(1, 2, -\frac{1}{2}) \oplus (1, 1, +1) \oplus (3, 2, + \frac{1}{6}) \oplus (\overline{3}, 1, -\frac{2}{3}) \oplus (\overline{3}, 1, +\frac{1}{3})$, and a complex scalar field in the representation $(1, 2, -\frac{1}{2})$. Here the last entry of each triplet gives the value of the $U(1)$ charge, known as $\it{hypercharge}$.

I am puzzled by the group representation $(1, 2, -\frac{1}{2}) \oplus (1, 1, +1) \oplus (3, 2, + \frac{1}{6}) \oplus (\overline{3}, 1, -\frac{2}{3}) \oplus (\overline{3}, 1, +\frac{1}{3})$. How does it come about? What are the steps (if any) to get this representation?
 A: Your text assumes you are familiar with the quantum number content of the elementary particle fermions, determined by the Millikan oil-drop experiment, structure functions of the light quarks, V-A structure of the weak currents, etc. These are experimental inputs and they come from out there, your world.
It helps you summarize the self-evident logic of their apparently diverse quantum numbers so you could write a compact QFT for them, that's all. I assume you seek an appreciation of the manifest logic involved.
It gives you the SU(3) color rep, singlet for leptons, color 3 for quarks, or color anti triplet for antiquarks. Likewise, their SU(2) weak isospin, vanishing for right handed singlets, and doublet for left-handers. (No separate 2-bars, of course, as SU(2) is pseudoreal.)
And, of course, mutatis mutandis for their CPT conjugates. You only have singlets and fundamental reps, since these are fundamental fermion building blocks of our world.
Thus, 


*

*(1,2,-1/2) , e.g. for $e _L$

*(1,1,1), e.g. for $\overline {e _R}$

*(3,2,1/6) e.g. for $u_L, d_L$

*($\bar 3$,1,-2/3) for $\overline{u_R}$  

*($\bar 3$,1,1/3) for $\overline{d_R}$.


The hypercharge in the third entry is dross -- an error-correction number, if you wish, given by $Y_W\equiv Q-T_3$, once you input the charge, in the  "minority usage", but actually modern  mainstream definition, so you might have to multiply it by 2 to agree with hidebound historical listings, like those linked here. It is the eigenvalue of U(1), as your particles are all singlets, of course, under it, and multiplies the B coupling charge of the fermion currents. The sooner you get used to its Golden Mnemonic, the better: It is the average charge of isomultiplets.
There is nothing more to it. Given these numbers you may completely, and concisely specify the fermion sector of the SM QFT.
