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I am quite new to representation theory and I reading Serre's Linear Representation of Finite Groups. In the first and second chapter, one trick he uses quite often is the substitution \begin{equation} h^0=\frac{1}{|G|}\sum_{t\in G}\rho^2_{t^{-1}}h\rho^1_{t}, \end{equation} where $\rho^j:G\to GL(V^j)$ are representations on $V^j$, $h:V_1\to V_2$ is linear, and $|G|$ is the order of the group. For instance he uses this when proving every representation is the direct sum of irreducible representations, and when proving Schur's lemma.

I wonder whether there is something behind this powerful trick. To me it seems like a method to gauge the torsion/ tension between the two representations, and then averaging over $G$, but I am not sure.

Thanks!

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Averaging works for any representation - it is a projection onto invariant vectors. In your case you see the space of linear maps $V_1\to V_2$ as a representation via $t\cdot h =\rho^2_t h \rho^1_{t^{-1}}$. –  user8268 Oct 16 '12 at 19:11

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Given an irreducible representations $\rho_1$ and $\rho_2$ of $G$, we can define the $G$-intertwiner $$P : V_1 \mapsto V_2, \qquad v \mapsto \frac{1}{|G|} \sum\limits_{g \in G} \rho_1(g^{-1}) h ( \rho_2(g) v)$$ onto the maximal invariant subspace of $h(V_1)$, whose irreducible components are equivalent to irreducible subrepresentation of $V_2$. Intertwiner are the canonical tool to compare $G$-reps.

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