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.

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