Trivial linear representation

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Definition

Let $G$ be a group, and $K$ be a field.

Then the map $\rho :G\to GL_{1}(K)$ sending each element of $G$ to the $1\times 1$ identity matrix is a linear representation. It is called the trivial representation.

This representation is often denoted $\mathbf {1}$.

Character

The character of this representation is $1$ on all elements of the group.