This question comes from [Srednicki's textbook "Quantum Field Theory"][1]. On page 532, the left-handed Weyl fields $\it{l}$ (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 $\it{l}$ 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 don't know how to calculate these products. I try to 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?
Besides, I don't know how to calculate the next two products. The second is a product of a representation of $SU(2)$ and a representationn of $U(1)$, while the third is the product of two representations of $U(1)$. In these cases, can we still use Young tableaux to do the calculations for the first entries? if yes, I don't know how. Or should we do the calculation in some other ways?





[1]: http://web.physics.ucsb.edu/~mark/qft.html