The trivial representation, the regular representation, and permutation representations from finite $G$-sets

Let $k$ be a field and let $G$ be a finite group.

The trivial representation of $G$ over $k$ is the one-dimensional representation on the vector space $k$ in which every $g\in G$ acts as the identity map.

The regular representation of $G$ over $k$ is the representation on the vector space $k[G]$ (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements) given by left multiplication by the basis units: $g\cdot x:=[g]x(g\in G,x\in k[G])\mathrm{.}$ Equivalently, on the basis vectors of $k[G]$, $g\cdot [h]=[gh]\mathrm{.}$ This uses the ring structure and unit property proved in The group ring $R[G]$ is a unital $R$-algebra with basis $G$, and each $g\in G$ is a unit of $R[G]$.

More generally, if $X$ is a finite left $G$-set (Left group actions, transitive actions, and faithful actions), the free $k$-module $k^{(X)}$ on $X$ (The free module on a set and its standard basis) becomes a representation by $g\cdot e_x:=e_{g\cdot x}(g\in G,x\in X)\mathrm{.}$ This is the permutation representation attached to the action of $G$ on $X$.

• The regular representation is the permutation representation of $G$ acting on itself by left translation.

• Because $G$ and $X$ are finite, these constructions are finite-dimensional.