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

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?

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.