Standard Model fermion generation

Definition

A Standard Model fermion generation is one copy of the quark and lepton multiplets carrying the following representations of $SU(2)_L\times SU(3)_C\times U(1)_Y$ :

\begin{array}{c|c} \text{field}&(SU(2),SU(3))_Y\\ \hline Q_L&(2,3)_{1/3}\\ L_L&(2,1)_{-1}\\ u_R&(1,3)_{4/3}\\ d_R&(1,3)_{-2/3}\\ e_R&(1,1)_{-2}\\ \nu_R&(1,1)_0\quad\text{if included.} \end{array}

Here $Q=T_3+Y/2$ . The minimal Standard Model omits $\nu_R$ ; adjoining it gives $16$ complex internal Weyl species in a generation. Antiparticles transform in the dual representations.

Left-handed Weyl convention

One may describe every field as left-handed by replacing right-handed fields with their charge conjugates. The same generation is then

(2,3)_{1/3}\oplus(2,1)_{-1}\oplus (1,\bar 3)_{-4/3}\oplus(1,\bar 3)_{2/3}\oplus (1,1)_2\oplus(1,1)_0.

Mixing this convention with the right-handed-field convention conjugates some color representations and reverses their hypercharges.

What “generation” records

The three observed generations have the same gauge representations but different masses and flavor-mixing data. The representation above records only internal gauge quantum numbers; it excludes momentum, spinor components, Yukawa couplings, and empirical flavor labels.

Exterior-algebra packaging

Including $\nu_R$ , one generation plus its antiparticles is a $32$ -dimensional representation isomorphic to $\Lambda\mathbb C^5$ restricted from $SU(5)$ . In the $E_7$ construction, three subspaces carry three copies of this $\mathfrak g_{\mathrm{SM}}$ -module. That is a mathematical multiplicity statement, not by itself a dynamical explanation of flavor.

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: §§2.3 and 3.1.
John C. Baez, “Three Generations in $E_7$ ,” 2026. arXiv record. Relevant: §§1 and 9–10.
David Tong, “The Standard Model,” lecture notes, 2017. arXiv record.