Knowlpedia

Theorem

E7 branching under A2 + A5

The adjoint E7 module branches under the mutually centralizing generation sl3 and standard sl6 as 8 + 35 + (3⊗15) + (3⊗15).

Statement

For the maximal mutually centralizing subalgebra

sl3gen⊕sl6SM⊂e7,\mathfrak{sl}_3^{\mathrm{gen}} \oplus\mathfrak{sl}_6^{\mathrm{SM}} \subset\mathfrak e_7,

the adjoint representation has the

133  ↓A2+A5=(8,1)⊕(1,35)⊕(3,15)⊕(3∗,15∗).\mathbf{133}\;\downarrow_{A_2+A_5} =(\mathbf8,\mathbf1) \oplus(\mathbf1,\mathbf{35}) \oplus(\mathbf3,\mathbf{15}) \oplus(\mathbf3^*,\mathbf{15}^*).

Equivalently, as a module for sl3gen⊕sl6SM\mathfrak{sl}_3^{\mathrm{gen}}\oplus\mathfrak{sl}_6^{\mathrm{SM}},

e7≅sl3gen⊕sl6SM⊕(3⊗15)⊕(3∗⊗15∗).\mathfrak e_7 \cong\mathfrak{sl}_3^{\mathrm{gen}} \oplus\mathfrak{sl}_6^{\mathrm{SM}} \oplus(\mathbf3\otimes\mathbf{15}) \oplus(\mathbf3^*\otimes\mathbf{15}^*).
What the decomposition means

This is a decomposition of the adjoint module and of the underlying . The first two terms together form the sl3gen⊕sl6SM\mathfrak{sl}_3^{\mathrm{gen}}\oplus\mathfrak{sl}_6^{\mathrm{SM}}. The two 4545-dimensional tensor-product summands are irreducible modules; they are not asserted to be Lie subalgebras.

Identification of the 15s

After choosing highest-weight conventions for , one may take

15=Λ2C6,15∗=Λ4C6.\mathbf{15}=\Lambda^2\mathbb C^6, \qquad \mathbf{15}^*=\Lambda^4\mathbb C^6.

Under the standard sl5\mathfrak{sl}_5, these restrict to complementary pairs among Λ1C5,…,Λ4C5\Lambda^1\mathbb C^5,\ldots,\Lambda^4\mathbb C^5, producing the three unlabeled generation modules without neutrino singlets.

Choice independence

Unlike a decomposition into individually labeled generation spaces, this branching rule requires only the chosen good embedded gSM\mathfrak g_{\mathrm{SM}}: its generation sl3\mathfrak{sl}_3 and standard sl6\mathfrak{sl}_6 are intrinsic centralizers. exchange 3↔3∗\mathbf3\leftrightarrow\mathbf3^* and 15↔15∗\mathbf{15}\leftrightarrow\mathbf{15}^*, so the naming of either dual pair is conventional while the paired decomposition is invariant.

References
  1. John C. Baez, “Three Generations in E7,” 2026, Theorem 11. arXiv:2608.06271.
  2. R. Slansky, “Group Theory for Unified Model Building,” Physics Reports 79 (1981), 1–128, Table 52. DOI record90092-2).
  3. E. B. Dynkin, “Semisimple Subalgebras of Semisimple Lie Algebras,” American Mathematical Society Translations, Series 2, 6 (1957), 111–244.