regular representation

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