 Trivial representation

The trivial representation sends every group element to the identity operator. A one-dimensional constant-coordinate line in a permutation representation is an example: relabelling positions leaves a uniform assignment unchanged. A trivial action on a higher-dimensional vector space has many invariant subspaces and is therefore not an irreducible representation.

In the context of representation theory, which studies how groups can be represented through matrices and linear transformations, the trivial representation is a fundamental concept. The **trivial representation** of a group \( G \) is the simplest way of mapping elements of \( G \) to linear transformations. In this representation, every element of the group is represented by the identity transformation.