Regular representation

The (left) regular representation of an algebra $A$ is the linear representation $L$ of $A$ on the vector space $E=A$ defined by the formula $L(a)b=ab$ for all $a,b\in A$. Similarly, the formula $R(a)b=ba$, $a,b\in a$, defines an (anti-) representation of $A$ on the space $E=A$, called the (right) regular representation of $A$. If $A$ is a topological algebra (with continuous multiplication in all the variables), then $L$ and $R$ are continuous representations. If $A$ is an algebra with a unit element or a semi-simple algebra, then its regular representations are faithful (cf. Faithful representation).

A (right) regular representation of a group $G$ is a linear representation $R$ of $G$ on a space $E$ of complex-valued functions on $G$, defined by the formula

$$(R(g)f)(g_1)=f(g_1g),\quad g,g_1\in G,\quad f\in E,$$

provided that $E$ separates the points of $G$ and has the property that the function $g_1\mapsto f(g_1g)$, $g_1\in G$, belongs to $E$ for all $f\in E$, $g\in G$. Similarly, the formula

$$(L(g)f)(g_1)=f(g^{-1}g_1),\quad g,g_1\in G,\quad f\in E,$$

defines a (left) regular representation of $G$ on $E$, where the function $g\mapsto f(g^{-1}g_1)$, $g_1\in G$, is assumed to belong to $E$ for all $g\in G$, $f\in E$. If $G$ is a topological group, then $E$ is often the space of continuous functions on $G$. If $G$ is locally compact, then the (right) regular representation of $G$ is the (right) regular representation of $G$ on the space $L_2(G)$ constructed by means of the right-invariant Haar measure on $G$; the regular representation of a locally compact group is a continuous unitary representation, and the left and right regular representations are unitarily equivalent.

For a finite group $G$, the action of $G$ on the group ring $\mathbb{C}G$ gives the regular representation of $G$. This representation contains a copy of each of the irreducible representations of $G$.