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Let $\rho:G\to\mbox{GL}(V)$ be a finite dimensional representation of a finite group $G$. We can assume the base field is $\mathbb{Q}$, but it doesn't really matter. Then we also obtain a representation $\eta:G\to\mbox{GL}(\mbox{End}(V))$ where $\eta(g)(T)=\rho(g)T\rho(g)^{-1}$, where we think of $\mbox{End}(V)$ as being a vector space in the obvious manner.

Can one say anything about $\eta$? Or has it been studied in the literature? For example, if we know the irreducible subrepresentations of $V$, can we say anything about the irreducible representations appearing in $\eta$? This may be well known, but I haven't been able to find many references.

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In fact one can! First, if it's all the same to you, let's assume that the base field is algebraically closed so we can say a bit more about representation theory. One should first note that this representation is just $V^* \otimes_{\mathbb{C}} V$ so you can completely describe its character. Second one even has an explicit description of the copy of the trivial representation that occurs in this representation! It is precisely $\text{End}_{\mathbb{C}[G]}(V)$ and it is easy to compute the dimension of this trivial component in terms of the number and multiplicity of irreps of $G$ that occur in $V$ (this last bit is why I wanted the field algebraically closed).

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  • $\begingroup$ Ok this is great! Thanks! $\endgroup$
    – rfauffar
    Jun 25, 2016 at 4:06
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    $\begingroup$ Yeah. It is usually the analysis of this representation (in particular the isotopic component of the trivial representation) that leads to the result on the semisimplicity of finite dimensional representations of finite groups, and many of the other basic results in representation theory of finite groups. $\endgroup$
    – Sempliner
    Jun 25, 2016 at 4:34

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