Georgi–Glashow SU(5) embedding

Core idea

The Georgi–Glashow $SU(5)$ homomorphism is

\Phi:U(1)\times SU(2)\times SU(3)\longrightarrow SU(5), \qquad \Phi(z,A,B)= \begin{pmatrix} z^3A&0\\ 0&z^{-2}B \end{pmatrix}.

Its image is $S(U(2)\times U(3))$ , and its kernel is

\{(z,z^{-3}I_2,z^2I_3):z^6=1\}\cong\mathbb Z_6.

It therefore induces an embedding

\overline\Phi: \bigl(U(1)\times SU(2)\times SU(3)\bigr)/\mathbb Z_6 \hookrightarrow SU(5).

Why the exponents are

3

and

-2

The scalar actions commute with $SU(2)\times SU(3)$ , while

(z^3)^2(z^{-2})^3=1

ensures determinant one. For $z=e^{i\theta}$ , differentiating the scalar part gives the compact Lie-algebra element

i\operatorname{diag}(3,3,-2,-2,-2).

The associated Hermitian weight matrix is $\operatorname{diag}(3,3,-2,-2,-2)$ ; after dividing by $3$ , it is the weak-hypercharge operator in the convention $Q=T_3+Y/2$ .

Terminology

The direct product does not embed injectively by $\Phi$ . “The Standard Model embedding in $SU(5)$ ” means the induced embedding of its $\mathbb Z_6$ quotient, equivalently the inclusion $S(U(2)\times U(3))\subset SU(5)$ .

Representation-theoretic role

Restricting the natural $SU(5)$ -action on $\bigwedge\mathbb C^5$ along $\Phi$ gives the exterior-algebra model of one Standard Model generation and its antiparticles.

Physics scope

Mathematically, this is a group homomorphism and representation extension. The original Georgi–Glashow model additionally interprets $SU(5)$ as a unified gauge symmetry and introduces dynamics and symmetry breaking; those claims are extra structure.

References

Howard Georgi and Sheldon L. Glashow, “Unity of All Elementary-Particle Forces,” Physical Review Letters 32 (1974), 438–441. DOI record.
John C. Baez and John Huerta, “The Algebra of Grand Unified Theories,” Bulletin of the American Mathematical Society 47 (2010), 483–552. arXiv record. Relevant: §3.1.
John C. Baez, “Three Generations in $E_7$ ,” 2026. arXiv record. Relevant: §§1 and 8.