SU(5), hypercharge, and the Standard Model subgroup

The $SU(5)$ example connects three questions that a useful group-theory calculation must keep together: which multiplets appear, what are their abelian charges, and what is the global subgroup? All three follow from a specified block map. A list such as $10\to3+6+1$ supplies none of the conjugation or hypercharge information needed to identify Standard Model fields.

Open the ten-dimensional example. The workbench keeps Dynkin labels and exact charges beside familiar dimension labels. This page concerns finite-dimensional internal representations; Lorentz spin, dynamics, and the choice of quantum gauge theory require additional data. The representation-theoretic setting is developed in Baez and Huerta, The Algebra of Grand Unified Theories, Section 3.1.

Fix the map before reading the charges

Put the three-dimensional color block first and the two-dimensional weak block second. The workbench’s primitive integer generator and homomorphism are

$$\begin{aligned} X&=\operatorname{diag}(2,2,2,-3,-3),\\ \iota_X(A,B,z)&=\operatorname{diag}(z^2A,z^{-3}B),\\ A&\in SU(3),\qquad B\in SU(2),\qquad z=e^{i\theta}. \end{aligned}$$

The parameter $\theta$ has period $2\pi$. We define physical hypercharge by

$$Y=-\frac X6 =\operatorname{diag}\left(-\frac13,-\frac13,-\frac13, \frac12,\frac12\right), \qquad Q_{\rm em}=T_3+Y.$$

This fixes both the sign and the factor of six. Some references instead call $2Y$ the hypercharge and write $Q_{\rm em}=T_3+Y/2$; their numbers must be converted before comparing a table.

The product-group workbench uses the integer charge

$$Q_Y=6Y=-X.$$

To express the same subgroup in that convention, invert the circle parameter:

$$\iota_{Q_Y}(A,B,z) =\operatorname{diag}(z^{-2}A,z^3B).$$

The “use these multiplets” workflow therefore changes the sign of the exported integer charge when transferring this $3+2$ branching to the Standard Model convention. This is a change of convention, not a conjugation of the nonabelian representations.

Fundamental and antifundamental

The defining vector splits directly into its two blocks:

$$\begin{aligned} \mathbf5&\longrightarrow (\mathbf3,\mathbf1)_{X=2} \oplus(\mathbf1,\mathbf2)_{X=-3}\\ &=(\mathbf3,\mathbf1)_{Y=-1/3} \oplus(\mathbf1,\mathbf2)_{Y=1/2}. \end{aligned}$$

Complex conjugation reverses both the complex representations and the abelian charges. The $SU(2)$ doublet is equivalent to its complex conjugate as an abstract representation, while the color triplet is not. Thus

$$\overline{\mathbf5}\longrightarrow (\overline{\mathbf3},\mathbf1)_{Y=1/3} \oplus(\mathbf1,\mathbf2)_{Y=-1/2}.$$

In a convention where every fermion is a left-handed Weyl field, these are the internal representations of $d^c$ and the lepton doublet $L$. The superscript $c$ denotes the charge-conjugate left-handed field associated with a usual right-handed fermion. It changes the color representation and hypercharge; it is not a second copy with the same charges.

A scalar in a $\mathbf5$ instead contains an electroweak doublet of hypercharge $1/2$ and a colored triplet of hypercharge $-1/3$. The branching identifies those components. It does not explain why their masses should differ or select a scalar potential.

Derive the ten without a memorized table

The irrep with Dynkin labels $(0,1,0,0)$ is

$$\mathbf{10}=\Lambda^2\mathbf5.$$

For vector spaces $V$ and $W$,

$$\Lambda^2(V\oplus W) =\Lambda^2V\oplus(V\otimes W)\oplus\Lambda^2W.$$

Apply this to $\mathbf5=(\mathbf3,\mathbf1)_{-1/3}\oplus(\mathbf1,\mathbf2)_{1/2}$. Charges add under tensor products, $\Lambda^2\mathbf3\cong\overline{\mathbf3}$ for $SU(3)$, and $\Lambda^2\mathbf2\cong\mathbf1$ for $SU(2)$. Therefore

$$\mathbf{10}\longrightarrow (\overline{\mathbf3},\mathbf1)_{-2/3} \oplus(\mathbf3,\mathbf2)_{1/6} \oplus(\mathbf1,\mathbf1)_1,$$

where subscripts now denote $Y$. The dimensions add as $3+6+1=10$, but the construction supplies much more: the conjugate color triplet, the weak doublet, and the three different charges are fixed together.

One conventional fermion generation is consequently organized as

$$\overline{\mathbf5}\oplus\mathbf{10}.$$

The assignments in the all-left-handed convention are:

Field	$SU(3)\times SU(2)$	$Y$	Integer $Q_Y=6Y$
$Q$	$(\mathbf3,\mathbf2)$	$1/6$	$1$
$u^c$	$(\overline{\mathbf3},\mathbf1)$	$-2/3$	$-4$
$d^c$	$(\overline{\mathbf3},\mathbf1)$	$1/3$	$2$
$L$	$(\mathbf1,\mathbf2)$	$-1/2$	$-3$
$e^c$	$(\mathbf1,\mathbf1)$	$1$	$6$

For a doublet, $T_3=\pm\tfrac12$. The quark doublet therefore has electric charges $2/3$ and $-1/3$, while the lepton doublet has charges $0$ and $-1$. A gauge-singlet left-handed field $\nu^c$ of hypercharge zero can be added as an $SU(5)$ singlet; it is not contained in the fifteen components above.

The adjoint and a possible breaking pattern

Use $\mathbf5\otimes\overline{\mathbf5}=\mathbf1\oplus\mathbf{24}$. The block-preserving endomorphisms supply the two adjoints and two singlets; subtracting the parent trace removes one singlet. The off-diagonal blocks supply a representation and its conjugate:

$$\begin{aligned} \mathbf{24}\longrightarrow{}& (\mathbf8,\mathbf1)_0 \oplus(\mathbf1,\mathbf3)_0 \oplus(\mathbf1,\mathbf1)_0\\ &\oplus(\mathbf3,\mathbf2)_{-5/6} \oplus(\overline{\mathbf3},\mathbf2)_{5/6}. \end{aligned}$$

The dimensions are $8+3+1+6+6=24$. The off-diagonal hypercharges follow from differences of the diagonal entries of $Y$, so their signs also check the chosen block ordering.

An adjoint scalar expectation value proportional to $X$ has stabilizer $S(U(3)\times U(2))$. In a gauge theory with such a vacuum, the twelve generators outside the stabilizer are broken; their vector fields belong to the two six-dimensional sectors above. The representation calculation alone does not establish that a chosen scalar potential has this vacuum, that the vacuum is stable, or what the resulting masses are.

The embedded subgroup is a quotient

With $\omega_3=e^{2\pi i/3}$, the integer-hypercharge map has kernel

$$\ker\iota_{Q_Y} =\left\langle (\omega_3\mathbf1_3,-\mathbf1_2,e^{i\pi/3}) \right\rangle\cong\mathbb Z_6.$$

Indeed, $z^{-2}\omega_3=1$ and $z^3(-1)=1$ for $z=e^{i\pi/3}$. Hence the subgroup actually embedded in $SU(5)$ is

$$S(U(3)\times U(2)) \cong\frac{SU(3)\times SU(2)\times U(1)_{Q_Y}}{\mathbb Z_6}.$$

If $t_3$ is color triality and $t_2$ is the $SU(2)$ highest weight modulo two, the generator acts on a multiplet by

$$\exp\!\left[\frac{2\pi i}{6}(2t_3+3t_2+Q_Y)\right].$$

Every multiplet in the table obeys $2t_3+3t_2+Q_Y=0\pmod6$. For example, the quark doublet gives $2+3+1=6$. This ties fractional hypercharge to the two nonabelian centers without treating a fractional number as an integer charge of a separately normalized circle.

There are two distinct conclusions. The stated $SU(5)$ embedding has the full $\mathbb Z_6$ kernel. By contrast, the observed local matter representations permit quotients by any subgroup of that kernel. Taking the direct product, or dividing by $\mathbb Z_2$, $\mathbb Z_3$, or $\mathbb Z_6$, changes global data while preserving these local multiplets. Matter compatibility alone does not prove an $SU(5)$ ultraviolet completion. Tong’s Line Operators in the Standard Model, Introduction and Section 3 explains why additional line-operator data matter.

The product-group calculator computes the common central kernel of the entered fields and tests a specified embedded quotient. Its Standard Model convention is precisely $Q_Y=6Y$ with the generator above.

Hypercharge normalization and couplings

Integer charge normalization and generator normalization serve different purposes. The former fixes a circle and its representation lattice; the latter fixes the invariant quadratic form used in a gauge kinetic term. In the defining representation,

$$\operatorname{tr}_{\mathbf5}Y^2 =3\left(\frac13\right)^2+2\left(\frac12\right)^2 =\frac56.$$

For the $SU(5)$ convention $\operatorname{tr}_{\mathbf5}(T_aT_b)=\tfrac12\delta_{ab}$, the normalized hypercharge generator is

$$T_Y=\sqrt{\frac35}\,Y.$$

Matching $g_5T_Y$ to the Standard Model term $g_YY$ gives

$$g_Y=\sqrt{\frac35}\,g_5, \qquad g_1\equiv\sqrt{\frac53}\,g_Y=g_5$$

at the matching scale in the simplest normalization convention. These algebraic factors do not assert equality of the measured low-energy couplings: running, the spectrum, and threshold effects belong to a separate physical calculation.

From multiplets to interactions

The new SU(N) tensor workbench can verify that $\mathbf{10}\otimes\mathbf{10}$ contains $\overline{\mathbf5}$ and $\mathbf{10}\otimes\overline{\mathbf5}$ contains $\mathbf5$. At the level of $SU(5)$ indices, candidate singlet contractions therefore include

$$\epsilon_{abcde}\,T_i^{ab}T_j^{cd}H^e, \qquad T_i^{ab}F_{j,a}(H^\dagger)_b,$$

where $T^{ab}$ is antisymmetric, $F_a$ is in $\overline{\mathbf5}$, and $H^a$ is a scalar in $\mathbf5$. Lorentz spinor indices and coupling constants have been suppressed. A complete interaction must restore them and enforce field statistics and any additional symmetries. The branching and tensor tools supply the representation content; they do not silently choose normalized invariant tensors or a flavor model.

A dependable workflow is to retain the parent highest weight, the $3+2$ map, and the relation $Q_Y=-X$ in the exported calculation; inspect the quotient condition; and only then add dynamics, masses, and interactions. The block-branching reference supplies the general derivation and verification conditions. Source checking and exact automated checks support this edition; independent human scientific review remains pending.