# How to decompose the representation of $\rm SU(5)$?

This question comes from Srednicki's textbook "Quantum Field Theory". On pages 514-515, it states:

Under the unbroken $$\rm SU(3)\times SU(2) \times U(1)$$ subgroup, the $$5$$ representation of $$\rm SU(5)$$ transforms as $$$$5 ~\rightarrow~ \left(3, 1, -\frac{1}{3}\right) \oplus \left(1, 2, +\frac{1}{2}\right) .\tag{84.12/97.2}$$$$

I wonder ─ how is this decomposition derived?

1. Actually the Lie group $$G~:=~SU(3)\times SU(2) \times U(1)$$ is not a subgroup of $$SU(5)$$. However the standard model gauge group $$G/\mathbb{Z}_6$$ is a subgroup of the GUT gauge group $$SU(5)$$, cf. e.g. this & this Phys.SE posts.
2. Here we will argue at the level of Lie algebras $$su(3)\oplus su(2) \oplus u(1)\subseteq su(5).$$ In detail, we identify $$su(5)$$ with anti-Hermitian traceless $$5\times 5$$ matrices; $$su(3)$$ with anti-Hermitian traceless $$3\times 3$$ block matrices in rows/columns 1,2,3; and $$su(2)$$ with the anti-Hermitian traceless $$2\times 2$$ block matrices in rows/columns 4,5; while $$u(1)$$ is generated by the diagonal traceless matrix $${\rm diag}(-2,-2,-2,3,3)$$ times an imaginary number.
3. The vectorspace $$V_5=V_3\oplus V_2$$ of the defining/fundamental representation $$\underline{\bf 5}$$ of $$su(5)$$ is decomposed into the defining/fundamental representation $$\underline{\bf 3}$$ of $$su(3)$$ in rows 1,2,3; and the defining/fundamental representation $$\underline{\bf 2}$$ of $$su(2)$$ in rows 4,5.
4. On the other hand, the first three rows $$V_3$$ are a singlet under $$su(2)$$; while the last two rows $$V_2$$ are a singlet under $$su(3)$$.
5. Also note that the generator of $$u(1)$$ has the same weak hypercharge/eigenvalue $$-2i$$ and $$3i$$ on $$V_3$$ and $$V_2$$, respectively. The overall normalization of the weak hypercharge depends on conventions/choice of the $$u(1)$$ generator.
6. Altogether, the decomposition of the $$su(5)$$ representation becomes $$\underline{\bf 5}~~\cong~~(\underline{\bf 3}\otimes\underline{\bf 1})_{-\frac{1}{3}}~~\oplus~~ (\underline{\bf 1}\otimes\underline{\bf 2})_{\frac{1}{2}}.$$