## Equivalent representations

Posted: February 8, 2011 in Representations of Finite Groups
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Definition. Two representations $\rho_i : G \longrightarrow \text{GL}(V_i), \ i = 1,2,$ are called equivalent if $V_1 \cong V_2$ as $\mathbb{C}[G]$-modules.

Remark 1. If $V_1 \cong V_2$ as $\mathbb{C}[G]$-modules, then obviously $V_1 \cong V_2$ as $\mathbb{C}$-modules, i.e. $\dim_{\mathbb{C}} V_1 = \dim_{\mathbb{C}} V_2.$ The converse is not necessarily true though. Try to find a counter-example!

Remark 2. Let’s see what exactly “equivalent representations” means. Suppose that

$\rho_i : G \longrightarrow \text{GL}(V_i), \ i = 1,2,$

are equivalent. So there exists a $\mathbb{C}[G]$-module isomorphism $\varphi :V_1 \longrightarrow V_2.$ Thus for all $g \in G, \ v \in V_1$ we have

$\varphi(gv)=g \varphi(v).$

But, by definition $gv = \rho_1(g)(v)$ and $g \varphi(v) = \rho_2(g) \varphi(v),$ because $\varphi(v) \in V_2.$ Hence

$\varphi \rho_1(g)(v)= \rho_2(g) \varphi(v),$

and so $\varphi \rho_1(g)= \rho_2(g) \varphi.$ Therefore, since $\varphi$ is an isomorphism, we get that for all $g \in G$

$\rho_2(g) = \varphi \rho_1(g) \varphi^{-1}. \ \ \ \ \ \ \ \ \ \ \ (*)$

Remark 3. It follows from $(*)$ that the degree one representations $\rho_1, \rho_2$ are equivalent iff they are equal. This is clear because, in terms of matrices, $\rho_i(g)$ and $\varphi$ are $1 \times 1$ matrices in this case and so  scalars. Thus $(*)$ gives us $\rho_2(g)=\rho_1(g),$ for all $g \in G.$