Deduction of weak boson hypercharge in Georgi–Glashow $SU(5)$ grand unified theory (GUT)

Question

In the Georgi–Glashow model, the extra 12 bosons $X_\mu^a$, $Y_\mu^a$ that appear have weak hypercharge $y = -\frac{5}{6}$. I want to know how this is deduced from the weak hypercharge generator

$$Y = \begin{pmatrix} -\frac{1}{3} & 0 & 0 & 0 & 0 \\ 0 & -\frac{1}{3} & 0 & 0 & 0 \\ 0 & 0 & -\frac{1}{3} & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2} \end{pmatrix}.$$

I know the fields $\psi$ have the hyperload as the eigenvalue of this operator, i.e,

$$Y \psi = y \psi.$$

I believe that to deduce this I have some function like

$$f(Y, X_{\mu}^a) = y X_{\mu}^a,$$ but I can't find it. Overall, for all bosons in this model, how are weak hypercharges deduced?

Answer 1

1. The $X$ and $Y$ gauge fields sit in the $\bar{\bf 2} \otimes {\bf 3}$ block of the adjoint representation ${\bf 24}\subset \bar{\bf 5} \otimes {\bf 5}$, cf. e.g. this related Phys.SE post.

2. We find their $U(1)$-charge $-\frac{1}{2}-\frac{1}{3}=-\frac{5}{6}$ by adding their $U(1)$-charges of the $\bar{\bf 5}$ and ${\bf 5}$ representations, cf. e.g. this related Phys.SE post.