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I've studied (on a graduation level) both Representation Theory and Galois Theory and it looks to me that one theory could benefit from the other. However when I asked my research supervisor about the relationship between both, he said there is little to none and all in a post-doc leve.

I would like to ask if anyone knows what are the bridges between both theories and if anyone could point me out some papers and articles, if they know about it.

Fernando Nazario
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    There are many connections. It depends on what you would like to do. This site has a few posts, which you can read first, e.g., this one. In general, Galois representations is a possible topic. – Dietrich Burde Jan 18 '21 at 12:24
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    Although the title of the question will remind number theorists of the theory of Galois representations and its applications in hard number-theoretic questions, it would perhaps be interesting to see a simple of example of a representation-theoretic application to Galois groups of down-to-earth finite-degree field extensions. (None come to my mind.) – Jeroen van der Meer Jan 18 '21 at 13:03
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    Galois descent for representations is an interesting question already for the case of $\mathbb{R}$ and $\mathbb{C}$, where it provides a principled way to recover real representations from complex ones. I don't know a place where this is discussed in detail but you can see, for example, here: https://en.wikipedia.org/wiki/Frobenius%E2%80%93Schur_indicator – Qiaochu Yuan Jan 18 '21 at 20:56
  • Thank you very much for the references! It will help me a lot! – Fernando Nazario Jan 20 '21 at 09:49

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