  Four-point functions are at the heart of many interesting physical processes.
A prime example is the light-by-light scattering amplitude, which plays an
important role in the calculation of hadronic contributions to the anomalous
magnetic moment of the muon. In the calculation of such quantities one faces
the challenge of finding a suitable and well-behaved basis of tensor structures
in coordinate and/or momentum space. Provided all (or many) of the external
legs represent similar particle content, a powerful tool to construct and
organize such bases is the permutation group S4. We introduce an efficient
notation for dealing with the irreducible multiplets of S4, and we highlight
the merits of this treatment by exemplifying four-point functions with
gauge-boson legs such as the four-gluon vertex and the light-by-light
scattering amplitude. The multiplet analysis is also useful for isolating the
important kinematic regions and the dynamical singularity content of such
amplitudes. Our analysis serves as a basis for future efficient calculations of
these and similar objects.
