regular representation

Given a group $G$ , the regular representation of $G$ over a field $K$ is the representation $\rho:G\longrightarrow\operatorname{GL}(K^{G})$ 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}(gg_{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 $G$ .