Generation module as an even exterior algebra

Statement

For $k=1,2,3$ , define the $32$ -dimensional subspace

V_k:= \bigoplus_{r\in\{\pm\beta_k\}\sqcup\Phi_k} (\mathfrak e_7)_r.

Then $V_k$ is an $\mathfrak{sl}_6^{\mathrm{SM}}$ -submodule of the adjoint $\mathfrak e_7$ -module, and

V_k\cong \Lambda^0\mathbb C^6 \oplus\Lambda^2\mathbb C^6 \oplus\Lambda^4\mathbb C^6 \oplus\Lambda^6\mathbb C^6 =\Lambda^{\mathrm{even}}\mathbb C^6,

the even exterior algebra of the defining module.

This is an isomorphism of modules and vector spaces. The subspace $V_k$ is not generally a Lie subalgebra of $\mathfrak e_7$ .

Restriction to sl5 and the Standard Model

For $\mathbb C^6\cong\mathbb C^5\oplus\mathbb C$ as $\mathfrak{sl}_5^{\mathrm{SM}}$ -modules, the direct-sum exterior-algebra formula gives

\Lambda^{\mathrm{even}}(\mathbb C^5\oplus\mathbb C) \cong \Lambda^{\mathrm{even}}\mathbb C^5 \oplus \Lambda^{\mathrm{odd}}\mathbb C^5 =\Lambda\mathbb C^5.

Consequently $V_k\cong\Lambda\mathbb C^5$ as an $\mathfrak{sl}_5^{\mathrm{SM}}$ -module, and its restriction to $\mathfrak g_{\mathrm{SM}}$ is the full one-generation Standard Model representation.

The two added singlets

The root spaces $(\mathfrak e_7)_{\pm\beta_k}$ commute with $\mathfrak{sl}_6^{\mathrm{SM}}$ , so they supply the trivial modules $\Lambda^0\mathbb C^6$ and $\Lambda^6\mathbb C^6$ . Under $\mathfrak g_{\mathrm{SM}}$ they model the right-handed neutrino gauge singlet and its antiparticle, completing the 30-dimensional module.

Dependence on choices and conventions

The spaces $V_k$ require a compatible Cartan subalgebra and a labeling of the three root lines. Exchanging $\beta_k$ with $-\beta_k$ preserves $V_k$ but swaps the two singlet root spaces and hence the conventional degree-zero/degree-six assignment. Particle and antiparticle labels are likewise conventional; the module isomorphism is the invariant statement.

References

John C. Baez, “Three Generations in E7,” 2026, Theorem 12. 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. arXiv:0904.1556.
Benjamin Nasmith, “An Exceptional Combinatorial Sequence and Standard Model Particles,” 2020. arXiv:2012.03933.