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Given a finite group representation $\rho:G\to GL_n(\mathbb C)$ one knows that the trivial representation $\mathbb 1$ is contained in $End(\rho)$.

Let $\rho'$ be the other summand, i.e., $\rho'$ is defined by $\rho'\oplus \mathbb 1=End(\rho)$. Some sources say that $\rho'$ is obtained by composing the projection on $PGL_n$ with the adjoint of $PGL_n$.

This may be obvious, but is there an easy way to see that?

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  • $\begingroup$ Schur's lemma says $End(\rho) = \mathbb C$, for irreducible $\rho$. Are you thinking of $\rho \otimes \check \rho = 1 \oplus Ad(\rho)$? $\endgroup$
    – Kimball
    Commented May 25, 2015 at 13:43
  • $\begingroup$ Your question is not clear at all! What do you mean by "the projection on ${\rm PGL}_n$"? "the adjoint of ${\rm PGL}_n$"? $\endgroup$
    – Paul Broussous
    Commented May 26, 2015 at 11:45

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