Specify the fields
An interaction is built from fields, with their internal representations, Lorentz indices, charges, and statistics. Counting singlets in a tensor product is the first step. The field content determines which tensor subspace is relevant.
The A₄ triplet example
Three distinguishable A₄ triplets admit two invariant multilinear contractions. The cubic polynomial in one real commuting scalar triplet instead lies in the symmetric cube, which contains one singlet. The decompositions make the effect of field statistics explicit:
What the workbench counts
The finite-group workbench computes exact representation multiplicities and invariant counts for tensor and symmetric powers of the real A₄ triplet. Symmetric powers describe homogeneous polynomials in one commuting triplet. Tensor powers describe one factor from each of several distinguishable triplets.
Beyond internal symmetry
Fermions require the combined Lorentz and internal index symmetries together with Grassmann signs. Additional gauge charges can forbid a contraction that is invariant under one factor. Derivative operators also require a treatment of integration by parts and equations of motion.
A useful next question
Change the degree from three to four. The single-triplet symmetric fourth power contains two singlets, while four distinguishable triplets contain seven. The result panel shows the full decomposition and a dimension check.
From channels to interactions
The A₄ scalar model constructs explicit invariant polynomials. The spinor guide explains why Lorentz contraction and field statistics must be combined.