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Can somebody give me concrete problems in finite group representation theory that can be solved using induced representations?

Yes, I understand how induced representations are defined, and I know how to compute their characters, and I know the reciprocity formula. But it bothers me that I have never once needed to used this concept to solve a concrete problem!

For example, maybe somebody can name a finite group whose character table is difficult to construct - until you start looking at representations induced from representations of its subgroups? (Note that these induced representations don't necessarily need to be irreducible to be useful - they can be direct sums of easy-to-find irreps and a hard-to-find irrep. See Jim's answer here.)

If you know any such examples, please let me know, and please don't reveal the full solutions because I would like to try to solve them myself!

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    $\begingroup$ The irreducible representations of symmetric groups are usually constructed as submodules of Young modules, which themselves are induced representations of trivial representations from Young subgroups. Of course, these Young modules can also be defined from scratch in a simpler way; but the "induced representation" point of view provides a shortcut to proving some of their properties. There are probably far better examples around. $\endgroup$
    – darij grinberg
    Mar 2, 2017 at 17:36
  • $\begingroup$ @darijgrinberg Do you mean "the irreps can also be defined from scratch..."? Or did I misunderstand? $\endgroup$
    – Kenny Wong
    Mar 2, 2017 at 17:56
  • $\begingroup$ No, I mean that the Young modules can be defined from scratch as permutation modules on the sets of Young tabloids. $\endgroup$
    – darij grinberg
    Mar 2, 2017 at 18:09
  • $\begingroup$ Yes, got it now! $\endgroup$
    – Kenny Wong
    Mar 2, 2017 at 18:16

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Nearly the only strategy available for computing the character table of most interesting groups is to work with induced representations from understandable subgroups. For example, try computing the character table of $SL_2(\mathbb{F}_q)$. There are very few permutation representations available (and note that permutation representations are also induced representations) and eventually you'll either give up or start inducing.

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  • $\begingroup$ Do you have any advice on how to pick the right subgroups to induce from? Or is it just pot luck? (I don't want the induced rep to be the direct sum of easy-to-guess irreps. Nor do I want the induced rep to be the direct sum of easy-to-guess irreps plus TWO difficult-to-guess irreps.) $\endgroup$
    – Kenny Wong
    Mar 2, 2017 at 20:58
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    $\begingroup$ Find large subgroups whose representation theory you understand, e.g. large abelian subgroups. It might be useful to keep Brauer's induction theorem in mind: en.wikipedia.org/wiki/… $\endgroup$
    – Qiaochu Yuan
    Mar 2, 2017 at 21:00

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