Conjugation representation
= Conjugation representation
If $G$ acts on a vector space $W$, it acts on $\operatorname{End}(W)$ by conjugation:
$$
g\cdot X=\rho(g)X\rho(g)^{-1}.
$$
Under $\operatorname{End}(W)\cong W\otimes W^*$, this is the <tensor product of group representations> $W\otimes W^*$.