Trivial linear representation

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This article describes a particular irreducible linear representation for the following group: [[{{{group}}}]]. The representation is unique up to equivalence of linear representations and is irreducible, at least over its original field of definition in characteristic zero. The representation may also be definable over other characteristics by reducing the matrices modulo that characteristic, though it may behave somewhat differently in these characteristics.
For more on the linear representation theory of the group, see [[linear representation theory of {{{group}}}]].

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.