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Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ have numerous combinatorial interpretations, including the hive model by Knutson and Tao, see here. For other root systems, there are also several combinatorial interpretations, notably the Berenstein and Zelevinsky interpretation in terms of integer points in polytopes, see here. In the root system A, these are closely related to hives (see KT paper and also here). Despite extensive googling I can't seem to find the answer to this:

Question: What are hives for root systems BCD?

What am I missing?

UPDATE As Gjergji helpfully writes in the comments, this was asked earlier in this MO question. I am not sure if the answers there resolve the problem (JiaRui Fei's paper comes close, perhaps). However, if there are new answers they should be posted there, so it's best to close this question.

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  • $\begingroup$ One of the spectacular applications of hives/honeycombs was the proof of the saturation conjecture, and I recall once hearing that the analog of the saturation conjecture is false in other types. Of course that’s not a definitive reason there couldn’t be hives in other types, but might start to point to something… $\endgroup$
    – Sam Hopkins
    Commented May 10, 2022 at 20:45
  • $\begingroup$ @SamHopkins Right. But I think you are getting it backwards. KT proof established a saturation property of hives polytopes for the root system A. If there was a "hive polytope for the root system C", for example, it would not satisfy that property. However, it might satisfy some weaker properties which would be interesting to investigate. If only we know what that hive polytope was... $\endgroup$
    – Igor Pak
    Commented May 11, 2022 at 6:24
  • $\begingroup$ There are Gelfand-Tsetlin type polytopes for other types, and they have a close connection to BZ-polytopes, and thus hives. Perhaps that's a place to start. $\endgroup$
    – Per Alexandersson
    Commented May 11, 2022 at 8:00
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    $\begingroup$ This question came up here a while ago: mathoverflow.net/questions/33712/… JiaRui Fei's paper "Tensor Product Multiplicities via Upper Cluster Algebras" mentioned there is a possible candidate to an answer. $\endgroup$
    – Gjergji Zaimi
    Commented May 12, 2022 at 20:16
  • $\begingroup$ I agree it's the same question. Thanks, @GjergjiZaimi -- although I googled extensively I didn't see it. Since the old one doesn't have an answer I am not sure if I should click on "Yes, this answers my question". I am ok with closing this question and hopefully we'll see more answer to the old question. $\endgroup$
    – Igor Pak
    Commented May 18, 2022 at 19:23

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