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If $V_\lambda$, $V_\mu$ and $V_\nu$ are irreducible representations of $GL_n$, the Littlewood-Richardson coefficient $c_{\lambda\mu}^\nu$ denotes the multiplicity of $V_\nu$ in the direct sum decomposition of the tensor product of $V_\lambda$ and $V_\nu$. Knutson and Tao proposed a ``Hive model" for Littlewood-Richardson coefficients in http://www.ams.org/journals/jams/1999-12-04/S0894-0347-99-00299-4/S0894-0347-99-00299-4.pdf

Is there an analogous model for such tensor product multiplicities for Lie groups of types B, C or D?

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The recent work of Goncharov-Shen gives a good generalization of the hive model for any reductive group $ G$. They show that $n$-fold tensor product multiplicities for $ G $ are counted by positive tropical integral points of the space $ G^\vee \setminus ( G^\vee / N)^n $. When $ G = GL_m$ and $n = 3$, this gives the Hive model. When $ G = GL_m $ and $ n = 4 $, this gives the octahedron recurrence.

Unfortunately, outside of type $ A$, it is hard to give a simple description of their set of positive tropical points. To do so requires some choices, and once one makes these choices, you end up with one of the Berenstein-Zelevinsky models which Allen mentioned.

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There are conjectural ones in the Berenstein-Zelevinsky paper referenced in that one. They have another paper with a general theorem, Tensor product multiplicities, canonical bases and totally positive varieties, that gives (many) polyhedral models for any Lie type.

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  • $\begingroup$ Hi Allen, Thank you very much. I think I heard that there may be a model for type B, where 3 hives are pasted together to form a Moebius strip, but otherwise the model is similar to the hive model for type A. I'd be very grateful if you could clarify if this is the case. $\endgroup$ – Hari Jul 30 '10 at 22:49
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There is some recent progress on this question. JiaRui Fei http://de.arxiv.org/pdf/1603.02521.pdf has given a generalization of the hive model for all semisimple Lie algebras, based on the theory of cluster algebras. In type A, his model specializes to the Knutson-Tao hive model.

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