Thirty-dimensional generation module in e7

Statement

For a labeled defining weight $w_k$ in the generation plane, let

\Phi_k=\{r\in\Phi(E_7):\pi(r)=\pm w_k\}, \qquad N_k:=\bigoplus_{r\in\Phi_k}(\mathfrak e_7)_r.

Then $N_k$ is a $30$ -dimensional $\mathfrak{sl}_6^{\mathrm{SM}}$ -submodule of the adjoint $\mathfrak e_7$ -module, and

N_k\cong \Lambda^2\mathbb C^6\oplus\Lambda^4\mathbb C^6

as $\mathfrak{sl}_6^{\mathrm{SM}}$ -modules, where the summands are exterior-power representations.

This is a module and vector-space decomposition. In general $N_k$ is not a Lie subalgebra of $\mathfrak e_7$ .

Restriction to the standard sl5

Using the defining-module splitting $\mathbb C^6\cong\mathbb C^5\oplus\mathbb C$ , restriction to $\mathfrak{sl}_5^{\mathrm{SM}}$ gives

N_k\cong \Lambda^1\mathbb C^5 \oplus\Lambda^2\mathbb C^5 \oplus\Lambda^3\mathbb C^5 \oplus\Lambda^4\mathbb C^5.

The formula follows from the exterior algebra of a direct sum and its degree decomposition.

Standard Model interpretation

On restriction to $\mathfrak g_{\mathrm{SM}}$ , this is the one-generation exterior-algebra representation with the degree-zero and degree-five singlets omitted. These two missing one-dimensional summands are the right-handed neutrino and its antiparticle, both gauge singlets. Adding the two generation-root spaces produces the 32-dimensional generation module.

Dependence on choices and conventions

The three spaces $N_k$ require a compatible Cartan subalgebra and a labeling of the three defining weights. Reversing highest-weight conventions exchanges $\Lambda^2\mathbb C^6$ with its dual $\Lambda^4\mathbb C^6$ ; their direct sum is unaffected. The particle/antiparticle and chirality labels additionally depend on the chosen Standard Model representation conventions.

References

John C. Baez, “Three Generations in E7,” 2026, Lemma 9 and Theorem 10. arXiv:2608.06271.
John C. Baez and John Huerta, “The Algebra of Grand Unified Theories,” Bulletin of the American Mathematical Society 47 (2010), 483–552, especially the exterior-algebra description of fermions. arXiv:0904.1556.