The decomposition of the curvature tensor of a (pseudo) riemmanian manifold into scalar+ traceless Ricci + Weyl (the latter into SD+ASD in dim=4) is an application of the representation theory of the orthogonal group. There are many more examples in differential geometry (eg the decomposition of the intrinsic torsion tensor of an almost hermitian manifold into 4 irreducibles etc).
Now you may object because the orthogonal group (say over R) is not a finite group, but Weyl showed that the theory of the tensor representations of the classical groups is intimately related to the representation theory of the symmetric group.
