Example 2

The (left) regular representation of G is the representation $\boldsymbol{\gamma }$ of G in K[G] defined by $\boldsymbol{\gamma }(g)(x) =gx$ for $g\in G$ and $x\in K[G]$. It corresponds to the left regular representation of the algebra K[G] (VIII, p. 375). Suppose that the group G is finite. The regular representation then has finite degree. For every element $g$ of G different from the identity, the matrix of left multiplication by $g$ in K[G] with respect to the canonical basis is the matrix of a permutation without fixed point. We therefore have

$|G|$ if $g$ is the identity element,

0 otherwise.

We define the right regular representation of G likewise. The biregular representation of G is the representation $\rho$ of $G\times G$ in K[G] defined by $\rho (g, g')(x) =gxg^{'-1}$ for $(g, g')\in G\times G$ and $x\in K[G]$.