What are singlet representations for the electroweak gauge group $SU(2)\times U(1)$?

Question

This question comes from Srednicki's textbook Quantum Field Theory. On page 532, the left-handed Weyl fields $\ell$ (a single lepton family, electron and its neutrino) and $\overline{e}$ are in the representations $(2, -\frac{1}{2})$ and $(1, +1)$ of $SU(2) \times U(1)$. It is stated:

We cannot write down a mass term involving $\ell$ and/or $\overline{e}$ because there is no gauge-group singlet contained in any of the products \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}), \\ (2, -\frac{1}{2}) \otimes (1, +1), \\ (1, +1) \otimes (1, +1) .\tag{88.4} \end{equation}

I calculate the first product as follows:

1. Using Young tableaux to do the calculation for the first entry in $SU(2)$ gives \begin{equation} 2 \otimes 2 = 1 \oplus 3 \end{equation}
2. For the second entry, I simply use addition \begin{equation} -\frac{1}{2} - \frac{1}{2} = -1 \end{equation}
3. Combining 1. and 2., I get the result \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}) = (1, -1) \oplus (3, -1) \end{equation} Is this correct? If yes, isn't $(1, -1)$ a singlet, which will refute the statement in the text?

Answer 1

1. Srednicki is merely trying to convey that terms (such as, e.g., a mass term) in the Lagrangian should be gauge-invariant under the electroweak gauge group $SU(2)\times U(1),$ i.e. belong to the trivial representation $(1,0).$ In particular, the three tensor products mention in eq. (88.4) do not contain the trivial representation $(1,0).$

2. OP's question seems to be spurred by the fact that all irreducible representations (over a complex vector space) of an abelian Lie group (such as, e.g., $U(1)$) are 1-dimensional. So shouldn't they all be called singlets? However, Srednicki seems to adapt the opposite convention that the trivial $U(1)$ irreducible representation is the only $U(1)$ singlet.