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Let $V_j$, $j = 1,2$ be finite dimensional representations of a group $G$. Show:

$\chi_{V_j}$ is a constant on each conjugacy class of $G$, where $\chi_{V_j}$ is the character of the representation.

I’ve just started with group theory and have a really hard time so I’d like someone to confirm what I did so far was correct:

Per definition: $\chi_{V_j} = Tr(\rho(g))$ where $\rho$ is the grouphomomorphism $G \rightarrow GL(V)$ which represents $G$. The conjugacy class of $G$ is defined as $\{ ghg^{-1} | g \in G \}$ so I’m just plugging in:

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$\chi_{V_j}(ghg^{-1}) = Tr(\rho(ghg^{-1}))$ which is the same as (because it’s a group hom.) $Tr(\rho(g) \rho(h) \rho(g^{-1}))$ and since $Tr(AB) = Tr(BA)$ we get:

$Tr(\rho(g) \rho(h) \rho(g^{-1})) = Tr(\rho(h) \rho(g^{-1}) \rho(g)) = Tr(\rho(h))$

Now since $h$ is a constant element of $G$ this is a constant function and I’m done.

Is this correct?

Thanks a lot in advance!

Cheers

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That is correct save one small point about terminology, which is probably a typo and not a misunderstanding:

You say that the conjugacy class of $G$ is defined as $\{ghg^{-1} \mid g \in G\}$. But the group as a whole does not have a conjugacy class. That is the conjugacy class of the element $h$.

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