Finite groups · workbench

Invariant couplings

Compare distinct fields with one identical commuting scalar.

Supported: A₄ triplet, degree 0–12

A₄ triplet powers1 invariant

\operatorname{Sym}^{3}(\mathbf3)=\mathbf1\oplus 3\cdot\mathbf3

Degree: 3
Complex dimension: 10
Trivial-irrep multiplicity: 1

A4 irreducible multiplicities
A₄ irrep	1	1′	1″	3
Multiplicity	1	0	0	3

Symmetric powers describe homogeneous polynomials in one identical commuting scalar triplet. Only the trivial irrep $\mathbf1$ counts as an A₄ invariant; $\mathbf1'$ and $\mathbf1''$ are nontrivial one-dimensional representations.

\begin{aligned}\dim[\mathbf3^{\otimes3}]^{A_4}&=2\\\dim[\operatorname{Sym}^{3}(\mathbf3)]^{A_4}&=1\end{aligned}

This counts internal A₄ invariants. A fermionic operator requires statistics on the complete field slots, including Lorentz and species indices. Additional charges, derivatives and operator relations require further data.

Method, conventions & sources

A4 character inner products; symmetric-power characters from the triplet eigenvalues. The irrep order is 1, 1-prime, 1-double-prime, 3.

Ordinary complex A4 representations. Only the trivial irrep 1 counts as an invariant; 1′ and 1″ carry nontrivial characters.

Calculation version: groups-design-1.0.0

How calculation links, versions, and exports work