Two remarks on this nice example: (1) The word 'orthonormal' doesn't have to be there; representations are a priori just linear actions; if they carry an invar

Two remarks on this nice example: (1) The word 'orthonormal' doesn't have to be there; representations are a priori just linear actions; if they carry an invariant inner product, that is a bonus ('unitary representation'). For compact (including finite) groups, the invariant inner product is no extra information, because we could pick any (not necessarily invariant) inner product and average it with respect to Haar measure to get an invariant one. This product is unique up to a scalar for an irreducible representation, but not in general (and the left-regular representation is never irreducible for a non-trivial finite group). (2) For people who are more used to cycle notation than the matrix representation of permutations, your permutation is (3 4). In roster notation, it's 1 2 3 4 1 2 4 3