trivial representation

The trivial representation is the simplest possible: the group does nothing at all. Every element acts as the identity on a one-dimensional space. It may seem too dull to mention, but it plays the role of the number 1 in the arithmetic of representations — it is the multiplicative unit for tensor products and the home of all group invariants.

Precisely, the trivial representation of G over a field k is the one-dimensional space k with action rho(g) = 1 (the identity scalar) for every g in G. Its character is the constant function chi(g) = 1, and it is automatically irreducible because a one-dimensional space has no proper nonzero subspaces. There is exactly one trivial representation up to isomorphism, and it exists for every group.

Its importance is structural. The subspace of vectors fixed by all of G inside any representation V — the invariants V^G — is the same as the multiplicity of the trivial representation in V, computable as the inner product (chi_V, chi_trivial). The trivial representation is also the unit object for the tensor product of representations: V tensor (trivial) is naturally isomorphic to V.