quantum field theory - Hypercharge for $U(1)$ in $SU(2)times U(1)$ model

I understand that the fundamental representation of $U(1)$ amounts to a multiplication by a phase factor, e.g. EM. I thought that when it is extended to higher dimensional representations, it would just become a phase factor times the identity matrix.

• 
  

  Can someone explain where the hypercharge comes into the $U(1)$ generator matrices in $SU(2)\times U(1)$ model, e.g. $Y = -(1/2) I$ in $(2, -1/2)$ representation?

  
  


• 
  

  I don't quite understand where the "$-1/2$" comes from. Where do all these hypercharges come from?

  
  


• 
  
  

  What is the logic behind choosing a particular value like $-1/2$?