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I am studying the problem of decomposing tensor products of irreps of $S_n$. As a non-expert, I was surprised to see that this is an open problem, or a the very least, one for which no satisfactory combinatorial answer has been given. Please correct me if I missed something in the literature here.

I was wondering whether the Okounkov-Vershik approach makes this problem easier or more transparent.

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    $\begingroup$ I believe Igor Pak has lots of insights and/or strong opinions why one should not expect a nice answer. These coefficients are known to be hard to compute (but, so are the Littlewood-Richardson coefficients and they do have a rule, so who knows...) $\endgroup$
    – Per Alexandersson
    Commented Aug 29 at 21:17
  • $\begingroup$ Roughly the Okounkov-Vershik approach, via the Jucys-Murphy elements, exploits how the simple modules of the symmetric group behave under restriction. There is no indication that tensor products behave well under restriction, so I wouldn't expect Okounkov-Vershik to help with the Kronecker coefficients. This said, I'd be very happy to be proved wrong. $\endgroup$
    – Andrew
    Commented Aug 30 at 4:01
  • $\begingroup$ @PerAlexandersson Yes, Pak explains his point of view in What is a combinatorial interpretation?. See Section 9. He conjectures that the Kronecker coefficients are even harder to compute than the Littlewood-Richardson coefficients. Specifically, the latter are in #P, but Pak conjectures that the Kronecker coefficients are not even in #P. $\endgroup$
    – Timothy Chow
    Commented Aug 30 at 11:49
  • $\begingroup$ I personally agree with Pak that we should not tacitly assume that there is no ignorabimus; i.e., that there must be a combinatorial interpretation, and we just have to work harder to find it. Whether Pak's proposal that #P is the right place to draw the line is IMO debatable, but if the Kronecker coefficients are hard for a complexity class larger than #P, then that would be good to know, and might not be that hard to prove. It would mean that any combinatorial interpretation would at least have to be "exotic". $\endgroup$
    – Timothy Chow
    Commented Aug 30 at 12:07

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Well the Okounkov-Vershik approach is close to 20 years old now, and it has been fairly well adopted by the community at large. So I won't say it could never help with the Kronecker coefficient problem, but if it were a silver bullet for solving it we would know by now.

Anyway here is in my mind what makes it hard to get at tensor products this way:

A group element $g$ acts on a tensor product by $g(u \otimes v) = gu \otimes gv$, but for a general element of the group algebra it is more complicated. In particular it is not so straightforward to figure out how the Jucys-Murphy elements act on a tensor product -- but analyzing how Jucys-Murphy elements act is an essential part of the Okounkov-Vershik approach.

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