Timeline for Does the Okounkov-Vershik approach to the representation theory of $S_n$ shed new light on the problem of computing Kronecker coefficients?
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6 events
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| Aug 30 at 12:07 | comment | added | Timothy Chow | 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". | |
| Aug 30 at 11:49 | comment | added | Timothy Chow | @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. | |
| Aug 30 at 4:01 | comment | added | Andrew | 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. | |
| Aug 29 at 21:17 | comment | added | Per Alexandersson | 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...) | |
| Aug 29 at 12:45 | answer | added | Nate | timeline score: 9 | |
| Aug 29 at 12:30 | history | asked | Andres Collinucci | CC BY-SA 4.0 |