# regular representation

Given a group $G$, the 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 $G$.

Title regular representation RegularRepresentation 2013-03-22 12:17:40 2013-03-22 12:17:40 djao (24) djao (24) 5 djao (24) Definition msc 20C99