A very common application of representation theory is the representation of the symmetric group. There are a lot to be said about the representation of the symmetric group.

I am wondering if the representation of a more exotic finite group $G$ has been studied deeply?

On the Wikipedia page of the representation of the symmetric group we can see that it has application in symmetric functions problems and in quantum mechanic. If you have an example of such a group $G$ are there applications of the representation of this group $G$?

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    $\begingroup$ Note that given representations of large enough symmetric groups, you obtain induced representations for any finite group, since every finite group is isomorphic to a subgroup of some symmetric group. Clearly this does not classify the representations for these groups, but it should at least give you some impression of the importance of this example. I don't have an answer to your question though. $\endgroup$ – Matt Samuel Dec 10 '18 at 18:10
  • $\begingroup$ Maybe you would consider this cheating, but the first case that comes to mind are finite cyclic groups. The (much simpler) theory is used usually under the name "discrete Fourier transform". $\endgroup$ – Pavel Čoupek Dec 10 '18 at 20:20

If $G$ is a finite group there is a bijection between conjugacy classes and irreducible representations but it's not clear how to do it in practice. The only case is for $S_n$ where everything is really explicit, and indexed by partitions of $n$.

However, it was a great advance to discover that one can realize such a bijection geometrically, by Springer theory. Representations of a general Weyl group can also be constructed using similar geometric methods, but in this case there is no natural bijection anymore (as Stephen mentionned in the comments).

The idea is that there is a geometric action of the Weyl group on the cohomology of the Springer fibers, i.e the fibers of the Springer resolution of the nilpotent cone. To my knowledge, people do not study intensively the representation theory of the Weyl groups today, but this was the starting point of a really fruitful theory which helped to understand a lot of algebras (for example the construction also works for universal enveloping algebras and their affine version) using some geometric constructions. A reference for that is the book by Chriss and Ginzburg, "Complex geometry and representation theory".

Also as mentioned in the comment by Tobias Kildetoft, variations and deformations of $\Bbb C[W]$ are still studied, for example Hecke algebras, rational Cherednik algebras, ...

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    $\begingroup$ The Weyl groups may not be that closely studied directly any longer. But their twisted analogues in the form of Hecke algebras certainly are, as well as various newer refined versions such as the Hecke category. $\endgroup$ – Tobias Kildetoft Dec 10 '18 at 19:35
  • $\begingroup$ @TobiasKildetoft : Thank you this is an excellent point, I should have added it. $\endgroup$ – Nicolas Hemelsoet Dec 10 '18 at 19:43
  • $\begingroup$ I am not aware of a bijection between conjugacy classes of elements of the Weyl group and representations. Springer theory indexes the irreducible representations of the Weyl group $W$ by pairs consisting of a conjugacy class of unipotent elements in a reductive group with the given Weyl group $W$ and a representation of the corresponding component group that appears in the permutation representation on the components of the Borel subgroups containing the given unipotent element. Is there a natural bijection between such pairs and conjugacy classes in the Weyl group $W$? $\endgroup$ – Stephen Dec 13 '18 at 14:47
  • $\begingroup$ @Stephen : Sorry for the delay, you are right, I had in mind the case of $GL_n$ but I'm not sure how it works in general. I'll let this comment for a bit so you can see it and I think I'll delete my answer after. $\endgroup$ – Nicolas Hemelsoet Dec 15 '18 at 17:16
  • $\begingroup$ Ok, thanks I saw it! I don't think you have to delete your answer. Anyway, even for the symmetric group the bijection between conjugacy classes and irreps is not particularly natural (and indeed there are really two ways to do it). It's really better to think of the two sets as being dual, IMO. $\endgroup$ – Stephen Dec 17 '18 at 20:31

The book Group Representations in Probability and Statistics, by Diaconis, is all about applications of group representations.


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