My question is about the reduction of a representation of a group $SU(5)$ to irreps of the subgroup $SU(3)\times SU(2) \times U(1)$.

For example the weights of the 10 dimensional representation of SU(5) are

One can identify the irreps of the subgroup by regrouping the dynkin labels into $((a_3 a_4) ,(a_1), a_2)$ such that (denoting $-1$ by $\bar{1}$):
$$ (1,1)_{Y} \rightarrow \left\{
\begin{array}{l l}
(0 0,0,1 )
\end{array} \right. $$

$$ (\overline{3},1)_{Y} \rightarrow \left\{
\begin{array}{l l}
(0 1,(0),\bar{1}) \\
(1 \bar{1},(0),\bar{1})\\
(\bar{1}0,(0),0)
\end{array} \right. $$

$$ (3,2)_{Y} \rightarrow \left\{
\begin{array}{l l}
(1 0,1,\bar{1}) \\
(\bar{1} 1,\bar{1},1)\\
(0\bar{1},\bar{1},1)\\
(1 0,\bar{1},0)\\
(\bar{1}1,1,0)\\
(0\bar{1},1,0)
\end{array} \right. $$

My problem is: how can I derive the $Y$ charge of the $U(1)$ factor for each of these from the Dynkin labels?

**Edit**

The metrictensor for SU(5) is thus

$$G= \frac{1}{5}\left(
\begin{array}{cccc}
4 & 3 & 2 & 1 \\
3 & 6 & 4 & 2 \\
2 & 4 & 6 & 3 \\
1 & 2 & 3 & 4
\end{array}
\right). $$

However in the reference, Slansky, on page 84 the same exercise is done but the axis have negative values...
$$\tilde{Y}^W = \frac{1}{3} [-2 \;1\, -1\; 2]. $$

How come they do not agree?

This post imported from StackExchange Physics at 2015-03-08 15:18 (UTC), posted by SE-user user41746