How Are SU(2) and U(1) Representations Combined in Particle Physics?

[lalo_u]

Well, i´m trying to understand this:

I´ve got a representation of $SU(2)_L\otimes U(1)_Y$ such that the left lepton doublets can be represented as (2, -1) and lepton singlets rights as (1, -2).

Then I can be left antiparticles bilinear representations as $(2,1)\times(2,1)$ or $(1,-2)\times(1,-2)$.

I wonder why, in the first case the possibilities are $(1,2)+(3,2)$, and in the second case is $(1,-4)$.

What kind of math operation has been done here?

 

[malawi_glenn]

Look up "Young tableux". You can also look up chapter 70 in the Quantum Field theory book by Srednikci , free draft here: http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf
Is also covered in any "group theory for physics" book.

In the case of $(2,1) \otimes (2,1)$ we have that the U(1) representations just add like normal numbers so 1+1 = 2. The tensor product of two SU(2) doublets becomes $2 \otimes 2 = 3 \oplus 1$ in other words one SU(2) triplet and one SU(2) singlet representation