# Why must $U(1)$ matrices commute with $SU(2) \times SU(3)$ matrices in embedding within $SU(5)$?

I'm a physicist taking a groups course.

I can believe that the direct sum of the fundmental representations for the $$SU(2)$$ and $$SU(3)$$ matrices will work as an embedding of the $$SU(2) \times SU(3)$$ subgroup within the fundamental representation of $$SU(5)$$ (since you have a 5x5 matrix composed of a 2x2 block and a 3x3 block so obviously they act on separate vector subspaces as in the direct product).

However it is often said that to embed $$SU(2) \times SU(3) \times U(1)$$ we can just also use a representation of 5x5 matrices for $$U(1)$$ which commutes with the above matrices. I don't understand how this commuting constraint means that the $$U(1)$$ part of the subgroup acts on a 'separate vector space' like the direct product suggests?

• $U(1)$ is just a phase factor, so just a number, so it commutes with everything no? Jan 6, 2022 at 22:05
• These $U(1)$ reps are diagonal matrices with the same phase on the SU(2) block and a same (other) phase on the SU(3) block (in such a way that the overall determinant of the matrix =1) Jan 6, 2022 at 22:06
• Related: physics.stackexchange.com/q/176544/2451 and links therein. Jan 6, 2022 at 22:10
• Could you help me understand the difference? Jan 8, 2022 at 0:24

So, a group element in the unbroken subgroup of SU(5) you are considering is $$\large e^{i (\theta \cdot \tau +3\alpha I_2 )~\oplus ~ i(\phi\cdot T -2\alpha I_3)} =e^{i (\theta \cdot \tau +3\alpha I_2) } ~\oplus ~ e^{i(\phi\cdot T -2\alpha I_3)},$$ where θ, φ, α are the SU(2), SU(3), and hypercharge "angle parameters", respectively (a triplet, octet, and singlet, respectively). Work out the exponential of the block matrices to see how the blocks in the exponent evaluate to a direct sum of blocks as well, perhaps counterintuitively, but pursuant to the direct sum homomorphism, preserving the block split. The first direct summand now acts on the 2×2 block, while the second on the 3×3 block.