For the Lie algebra $\mathfrak{so}_{2n+1}$ where can I find a description of the fusion rules of it fundamental representations? In more detail: For $\pi_i$ and $\pi_j$ two fundamental weights of $\mathfrak{so}_{2n+1}$, $V_{\pi_i}$ and $V_{\pi_j}$ the corresponding irreducible representations, where can I find an explicit formula for determining the fundamental representations appearing on the RHS of the identity $$ V_{\pi_i} \otimes V_{\pi_j} \simeq V_{\pi_{i_1}} \oplus \dotsb \oplus V_{\pi_{i_m}}. $$ For the case of $\mathfrak{sl}_n$ there is a well-known approach using Young diagrams. I would like to see such an approach for the case of $\mathfrak{so}_{2n+1}$.
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$\begingroup$ There are various notions of "orthogonal tableaux" but they are all quite complicated (symplectic is usually easier). But "Littelmann paths" (see "A generalization of the Littlewood-Richardson rule" by Peter Littelmann, doi.org/10.1016/0021-8693(90)90086-4) works to give a tensor product decomposition rule in all types. $\endgroup$– Sam HopkinsApr 15, 2022 at 14:12
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$\begingroup$ See also en.wikipedia.org/wiki/Littelmann_path_model which compares the path model to other approaches like crystals and standard monomials. $\endgroup$– Sam HopkinsApr 15, 2022 at 14:14
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$\begingroup$ If you need to calculate specific examples you can also use the program LiE (wwwmathlabo.univ-poitiers.fr/~maavl/LiE). It normally has a web interface but that seems to be down at the moment. $\endgroup$– CallumApr 16, 2022 at 9:52
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$\begingroup$ Kwon has an explicit combinatorial rule, but I can answer properly only tomorrow. $\endgroup$– Martin RubeyApr 18, 2022 at 9:28
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