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I am studying representation theory. I met a question which I have no clue to solve, could someone help me?

the question is: Let $f$ be a representation from G → GL(V). The $Hom_G(f,f)$ is the set of intertwining operator from V to V. then prove the following:

  1. if $f$ is direct sum of $f_1$ and $f_2$,where $f_i$ is irreducible. and $f_1$ is isomorphic to $f_2$, then show $Hom_G(f,f)$ is isomorphic to $Mat_{2×2}(C)$

My thought is that if $f$ is direct sum of $f_1$ and $f_2$, then $f$ can be written as a 2 by 2 block matrix. Since T ∈ $Hom_G(f,f)$ is intertwining operator, then $Tf=fT$. which gives that T must be a 2 by 2 block matrix...so $Hom_G(f,f)$ is isomorphic to $Mat_{2×2}(C)$? I am not quite sure if my thoughts are correct?

Can someone help me with this? thanks!


Also there are some more subquestions which I am already solved (for someone who may be interested, the solved questions are as follows)

  1. $Hom_G(f,f)$ is an algebra

  2. if f is now isomorphic to the direct sum of $m_1f_1$, $m_2f_2$,... $m_nf_n$,then find the dimension $Hom_G(f,f)$

  3. if $f$ is direct sum of $f_1$ and $f_2$,where $f_i$ is irreducible. And $f_1$ is NOT isomorphic to $f_2$, then prove $Hom_G(f,f)$ is isomorphic to $C×C$ (with coordinate wise multiplication). Where C standards for complex number. $hint$ $use$ $schurs's$ $lemma$

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    $\begingroup$ So I assume the $f_i$ are assumed to be simple? Anyway, there are a lot of individual questions here, so you need to narrow it down to what you are actually having trouble with (the first one really ought to be almost trivial). $\endgroup$ – Tobias Kildetoft Oct 16 '16 at 18:48
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    $\begingroup$ Related : math.stackexchange.com/questions/1952543/… $\endgroup$ – Arnaud D. Oct 16 '16 at 19:08
  • $\begingroup$ @ArnaudD. Thanks! this really helps! $\endgroup$ – user368131 Oct 16 '16 at 19:19
  • $\begingroup$ @TobiasKildetoftThank you for your reply. I fi is irreducible...And i can manage to solve the first question. Probably if you can help me with the second question, I might be able to solve the rest of them. thanks! $\endgroup$ – user368131 Oct 16 '16 at 20:15
  • $\begingroup$ @ArnaudD. Hi, I have a thoughts about the questions about $Mat_{2×2}(C)$. Could you help me to check if it correct? I really appreciate your time and patience! $\endgroup$ – user368131 Oct 16 '16 at 21:29

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