Highest weights in a physics calculation

A multiplet is specified by its representation and its group. A dimension is useful shorthand once those choices are fixed. This guide uses highest weights to keep them explicit while answering four practical questions: how many components does the multiplet have, what are its quadratic invariants, is it of real or complex type, and does it descend to the intended global group?

Prerequisites: a Lie algebra, a representation, and the distinction between a group and its covering group. The algebras and global groups guide introduces the last distinction.

Goal: read and reproduce a highest-weight result, including the conventions needed to use it in a field-theory calculation.

Open the highest-weight workbench

1. Start with the domain

For a compact simply connected group with simple Lie algebra, every dominant integral weight

$$\lambda=a_1\omega_1+\cdots+a_r\omega_r, \qquad a_i\in\mathbb Z_{\geq0},$$

labels one finite-dimensional complex irrep. The same labels specify an irrep of its complexified Lie algebra. A quotient of the group admits only the representations on which its quotient kernel acts trivially. Conrad–Landesman, Lectures 27–28.

This is why the workbench begins with a compact simply connected group and states a separate descent result. For type $B_3$, the domain is $\operatorname{Spin}(7)$. Its irrep $(0,0,1)$ has complex dimension $8$ and is a spinor; it is not an ordinary representation of $SO(7)$. The SageMath highest-weight tutorial uses this same example.

2. Keep the labels when recording a dimension

For $SU(3)$ the labels are $(p,q)$, with

$$(p,q)^*=(q,p).$$

The representations $(1,0)$ and $(0,1)$ both have dimension $3$, but they are conjugate and inequivalent. Likewise, the two $10$-dimensional irreps are $(3,0)$ and $(0,3)$. A decomposition written only as a list of dimensions can lose this information.

For $SU(5)$, the fundamental weights give dimensions $5,10,10,5$. The second fundamental weight is the antisymmetric two-index representation:

$$V_{\omega_2}\cong\Lambda^2\mathbf5, \qquad \dim V_{\omega_2}=\binom52=10.$$

Its dual has highest weight $\omega_3$. Keeping the four labels distinguishes $\mathbf{10}$ from $\overline{\mathbf{10}}$ without relying on a particular typography. This is the type-$A$ exterior-power construction described in Etingof, §28.3.

3. Compute a useful set of invariants together

Choose type $A$, rank $2$, and labels $(1,1)$:

$$\dim V_{(1,1)}=8,\qquad C_2=3,\qquad T=3.$$

The calculation uses long roots of squared length $2$ and

$$C_2(\lambda)=\frac{(\lambda,\lambda+2\rho)}{2}, \qquad T(R)\,\dim\mathfrak g=C_2(R)\,\dim R.$$

This profile gives $T(\mathbf N)=\tfrac12$ for $SU(N)$. Compare normalization profiles before comparing Casimir or index values from another table. The dimension is unchanged by such a normalization choice.

The Weyl formula reference derives these values and records the root numbering. The workbench exposes the Weyl factors and Cartan matrix so the calculation can be checked from the highest weight.

Quadratic indices can enter field-theory loop calculations. They do not include the number of field species, their spacetime spin, or whether a field is taken as real or complex. Those additional choices belong to the physical setup.

4. Test global descent independently of reality

Consider the $SU(3)$ representation $(3,0)$. Direct substitution gives

$$\dim(3,0)=10,\qquad C_2(3,0)=6,\qquad T(3,0)=\frac{15}{2}.$$

The central generator $e^{2\pi i/3}I$ acts by the phase determined by

$$t=p+2q\pmod3.$$

Here $t=0$. The representation therefore descends to $PSU(3)=SU(3)/\mathbb Z_3$. Its conjugate is still $(0,3)$, so the representation remains of complex type. This example separates two frequently conflated tests: trivial center action and an invariant real structure. The center criterion follows from the lattice description in Conrad’s center notes.

Reproduce the $(3,0)$ calculation.

For an entire matter sector, use the matter and global forms tool. It checks the common central kernel of the specified $SU(N)$ multiplets. An admissible quotient is a constraint on the global gauge group; further choices of line operators and other theory data can still be relevant.

5. Interpret spinors with the covering group intact

For $D_5$ the simply connected compact group is $\operatorname{Spin}(10)$. In the site’s node order, the two branch ends, nodes $4$ and $5$, carry the half-spin weights:

$$(0,0,0,1,0),\qquad (0,0,0,0,1).$$

They have dimension $16$ and are conjugate complex representations. The vector $(1,0,0,0,0)$ has dimension $10$ and real type. These are exact evaluations of the displayed root data.

There are three different relevant global statements:

Representation	On $\operatorname{Spin}(10)$	On $SO(10)$	On $PSO(10)$
Vector $\mathbf{10}$	Yes	Yes	No
Either half-spin $\mathbf{16}$	Yes	No	No
Adjoint $\mathbf{45}$	Yes	Yes	Yes

The center of $\operatorname{Spin}(10)$ is $\mathbb Z_4$. The vector kills its order-two subgroup; a half-spin central character has order four; the adjoint kills the whole center. This uses the distinction between the spin cover, the orthogonal quotient, and the adjoint quotient in Conrad, pp. 2–4.

The workbench’s adjoint descent field refers to the last column. Read the actual global group before interpreting that result as a statement about an orthogonal group.

6. Use exceptional groups without hiding the conventions

The following are exact workbench examples. All highest weights refer to the node numbering in the formula reference; $E_6$ and $E_7$ denote their compact simply connected forms.

Group	Highest weight	Dimension	$C_2$	$T$	Type
$G_2$	$\omega_1$	$7$	$2$	$1$	Real
$F_4$	$\omega_4$	$26$	$6$	$3$	Real
$E_6$	$\omega_1$	$27$	$26/3$	$3$	Complex
$E_7$	$\omega_7$	$56$	$57/4$	$6$	Quaternionic
$E_8$	$\omega_8$	$248$	$30$	$30$	Real

The root realizations are documented in the primary SageMath type-E, type-F, and type-G references. The table’s quadratic numbers follow from the explicit weight metric and the declared normalization.

“Quaternionic” describes an invariant antilinear structure whose square is $-1$. It does not mean that the displayed complex dimension is to be halved when counting components of an arbitrary field. The field’s reality condition and spacetime representation require their own specification. For self-dual irreps, the underlying indicator is given by Adams, Lemma 5.3.

7. Move from one multiplet to a calculation

Once the labels and global group are fixed, three different operations become available:

• Tensor product: ask which irreps occur when two multiplets are combined. The SU(3) tensor workbench keeps multiplicities and conjugate labels.
• Restriction: choose a specific group map and decompose under it. The SU(3) branching workbench distinguishes the regular inclusion from the principal homomorphism.
• Global-form compatibility: ask which central subgroup acts trivially on all chosen matter. The matter workbench retains the individual charges.

A singlet in a tensor product gives a possible internal contraction. Identical-field statistics, Lorentz indices, derivative relations, and other charges can remove or identify physical operators. Those assumptions should accompany any invariant count.

Checks worth doing before citing a result

Keep the domain, coefficient field, Dynkin labels, numbering, and normalization with the result. Check whether the word “dimension” refers to the representation or the group, and whether “real” refers to a real form of the algebra or the type of a complex representation. Confirm the quotient kernel separately.

The current highest-weight engine covers 32 finite root systems through rank eight, with labels $0$–$30$. Its output is exact within that scope and includes versioned conventions and source references. A formula-supported parameter range is different from an exhaustive library of matrices or all noncompact unitary representations. Reproduction and citation guidance.

Next: SU(3) branching maps or finite-group tensor products.