Standard Model exterior-algebra representation

Core idea

Let $E=\mathbb C^2$ and $C=\mathbb C^3$ , and let

\Phi(z,A,B)=\operatorname{diag}(z^3A,z^{-2}B)\in SU(E\oplus C)\cong SU(5).

The Standard Model exterior-algebra representation is the restriction along $\Phi$ of the natural $SU(5)$ -action on

\bigwedge(E\oplus C)=\bigoplus_{k=0}^{5}\bigwedge^k\mathbb C^5.

It has complex dimension $2^5=32$ and is isomorphic to the internal-symmetry representation on one fermion generation together with its antiparticles, when a gauge-singlet right-handed neutrino is included. Lorentz-spin degrees of freedom are not part of this identification.

Decomposition into Standard Model multiplets

Write $(d_2,d_3)_Y$ for an $SU(2)\times SU(3)$ multiplet with weak hypercharge $Y$ , using $Q=T_3+Y/2$ . Since

E=(2,1)_1,\qquad C=(1,3)_{-2/3},

the exterior degrees decompose as

\begin{aligned} \Lambda^0 &: (1,1)_0,\\ \Lambda^1 &: (2,1)_1\oplus(1,3)_{-2/3},\\ \Lambda^2 &: (1,1)_2\oplus(2,3)_{1/3}\oplus(1,\bar 3)_{-4/3},\\ \Lambda^3 &: (1,1)_{-2}\oplus(2,\bar 3)_{-1/3}\oplus(1,3)_{4/3},\\ \Lambda^4 &: (2,1)_{-1}\oplus(1,\bar 3)_{2/3},\\ \Lambda^5 &: (1,1)_0. \end{aligned}

The summands in complementary degrees are dual, as follows from the $SU(5)$ -invariant volume form.

Role of the two trivial summands

The degree-zero and degree-five lines are trivial representations. They are identified with a right-handed neutrino and its antiparticle, in one order or the other. Omitting a right-handed neutrino removes these two lines and leaves a $30$ -dimensional particle-plus-antiparticle representation.

Scope and relation to

E_7

This is an isomorphism of complex representations. It neither makes wedge multiplication a physical product of particles nor includes their Lorentz-spin degrees. In the $E_7$ construction, three $32$ -dimensional subspaces each restrict to this same $\mathfrak g_{\mathrm{SM}}$ -representation. Each arises as $\Lambda^{\mathrm{even}}\mathbb C^6$ for an intermediate $\mathfrak{sl}_6$ , then restricts along $\mathbb C^6=\mathbb C^5\oplus\mathbb C$ to $\Lambda\mathbb C^5$ .

References

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 and Table 4.
John C. Baez, “Three Generations in $E_7$ ,” 2026. arXiv record. Relevant: §§9–10, especially Theorem 12.