Finite symmetries: from characters to interactions

Follow one symmetry action all the way to a coupling you can check. The running example is $S_3$ permuting three scalar components subject to $x_1+x_2+x_3=0$. This two-dimensional multiplet shows how a character table encodes a decomposition, and why identical fields require a symmetry projection before counting interactions.

You need linear algebra and tensor products. The representation calculations use ordinary complex representations in characteristic zero; the scalar example has a real form. Modular representations, antiunitary transformations, and projective representations require additional data and are outside this path’s calculator contract.

1. Specify the action

Read groups, actions, and representations, then what “all irreducibles” means. Distinguish the abstract group from its action on a field space and separate irreducibility from faithfulness.

Carry forward: the action $\sigma\cdot e_i=e_{\sigma(i)}$ on $\mathbb C^3$ has an invariant line and the invariant plane $W=\{x_1+x_2+x_3=0\}$. The permutation character counts fixed coordinates; subtracting the line gives $(2,0,-1)$ in the class order identity, transposition, three-cycle. The S3 record fixes the object and the standard-irrep record fixes the category and character.

Checkpoint: why does the existence of three conjugacy classes establish a finite number of ordinary complex irreps, but not provide their matrices? The character is a basis-independent trace; matrices need an additional realization and basis.

2. Read the table and its reality data

Read characters and reality types. Keep class sizes, class ordering, and the power map $g\mapsto g^2$ beside the character rows. Real-valued characters alone do not imply real type.

Carry forward: $W$ is of real type. The $Q_8$ doublet has a real-valued character but quaternionic type. You will calculate both indicators in the exercise below. These distinctions are the ordinary-complex definitions in Etingof et al., §5.1; changing the coefficient field is a separate operation.

3. Choose the correct product space

Read from characters to physical couplings, especially its comparison of distinguishable factors with identical fields. For two independently declared scalar multiplets, start with $W\otimes W$. For a homogeneous quadratic polynomial of one commuting multiplet, use $\operatorname{Sym}^2(W^*)$. For this real-type example, the dual is isomorphic to $W$.

Carry forward: the tensor square decomposes as $\mathbf1\oplus\mathrm{sgn}\oplus W$, while the symmetric square is $\mathbf1\oplus W$ and the exterior square is $\mathrm{sgn}$. The sign representation is one-dimensional but is not the trivial representation.

Checkpoint: identifying a singlet in a tensor product is not sufficient to show that it survives the statistics projection. The quaternionic doublet will provide a case in which it does not.

4. Derive, calculate, and retain the answer

Complete the finite-character exercise. It supplies the full $S_3$ table, square map, intermediate character vectors, and solutions, followed by the $Q_8$ comparison. Work it on paper before using the finite-character workbench.

Carry forward: one commuting $S_3$ doublet has one quadratic invariant. On the real plane it can be written

$$q=x_1^2+x_2^2+x_3^2 =2(x_1^2+x_1x_2+x_2^2),\qquad x_3=-x_1-x_2.$$

Permuting the three coordinates leaves $q$ fixed. Its coefficient can enter a scalar potential, while the character calculation supplies the dimension of the invariant space without choosing this particular normalization.

Inspect the result’s decomposition and dimension checks, then save the calculation as JSON and copy its calculation link. Reopening that link recomputes the result from the recorded inputs; this workbench has no JSON import control. The reproducibility guide explains which input, convention, data, and algorithm identifiers to retain. Exported values are not a replacement for recomputation.

5. Interpret the coupling in a physical model

Read finite internal symmetries in physical models. Specify whether the fields are real or complex, whether a conjugate appears, and which other charges and spacetime indices are present. An internal singlet is a kinematic permission; symmetry alone fixes neither its coefficient nor the dynamics.

Carry forward: an identical commuting complex $Q_8$ doublet has no holomorphic quadratic invariant, even though its unrestricted tensor square contains a singlet. A Hermitian term $\phi^\dagger\phi$ is allowed and uses a different product, $V^*\otimes V$. For fermions, read fields, Lorentz symmetry, and statistics before choosing an exterior power: exchange acts on the full field space.

6. Locate the result within the larger subject

Use finite-character formulas for repeat calculations and finite-character methods for exact arithmetic and power-map conventions. Browse the finite-symmetry collection for other small groups and compare with the finite-simple classification. The latter classifies simple groups, not every group action or a universal library of representation matrices.

The current character workbench supports $C_3,S_3,D_8,Q_8,A_4,S_4,A_5$, with $D_8$ denoting order eight. Its bounds are base complex dimension at most $60$, input irrep multiplicities at most $12$, and power degree $0$–$12$. These are computational limits, not boundaries of representation theory. Coverage distinguishes curated records and executable operations. For ordinary and modular character-table databases beyond this selection, start with the external-resource guide.

At the end of this path you should be able to name the category, recover a multiplicity with the correct class measure, test reality using the square map, and explain why the physical field space may change the answer. Continue to Lie groups and topology when your symmetry is continuous or the global group becomes part of the question.