Given a group $G$ , the regular representation of $G$ over a field $K$ is the representation $\rho: G \longrightarrow \GL(K^G)$ whose underlying vector space $K^G$ is the $K$ -vector space of formal linear combinations of elements of $G$ , defined by $$ \rho(g)\left(\sum_{i=1}^n k_i g_i\right) := \sum_{i=1}^n k_i (g g_i) $$ for $k_i \in K$ , $g, g_i \in G$ .

Equivalently, the regular representation is the induced representation on $G$ of the trivial representation on the subgroup $\{1\}$ of $G$ .