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When tensoring finite dimensional representations of the Lie algebra ${\frak sl}_n$, we have an explicit algorithm given in terms of Young diagrams. See Section 4 of this paper.

Do there exist similar pictures for the $B$ and $D$ series? I am specifically interested in simplest case, where one of the irreducible representations being tensored is the fundamental representation.

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    $\begingroup$ Are you just asking about a Type B Littlewood-Richardson rule? $\endgroup$ Sep 23, 2019 at 19:18
  • $\begingroup$ I guess so. Specifically L-R for tensoring by the first fundamental representation. $\endgroup$ Sep 24, 2019 at 14:50

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Littelmann used standard monomial theory to give a unified Littlewood–Richardson rule for the simple reductive algebraic groups of types $A$, $B$, $C$ and $D$ (and some others) in which the coefficients enumerate certain generalized standard tableaux. See (a) in the theorem on page 346 of Littlemann's paper, A generalization of the Littlewood-Richardson rule., J. Alg. 130 (1990) 328–368.

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  • $\begingroup$ Thanks for the reference! For the B-series, when tensoring by the fundamental representation, does this admit a nice pictorial presentation? $\endgroup$ Sep 26, 2019 at 12:54
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I am not sure this answers your question, but in type $A$, you have the notion of crystals, and tensor products of crystals. Connected components of crystal graphs correspond exactly to irreducible representations (Schur functions).

In type $B$, there is a similar story - there is the notion of so called queer crystals, which also has a tensor product rule. See for example these slides by A. Schilling. Another reference is this paper. The connected components here give the Schur's P functions.

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    $\begingroup$ Schur $P$ functions are related to the projective representation theory of the symmetric group, and to the cohomology of the orthogonal/Lagrangian Grassmannian, but are they related to the representation theory of the Type B Lie algebra? I'm not so sure about that.... I think Type B Lie algebra crystals are simpler than queer crystals (which ultimately come from the queer Lie superalgebra) $\endgroup$ Sep 25, 2019 at 17:55
  • $\begingroup$ @SamHopkins Ah, do you have a reference for type B Lie algebra crystals? What are the underlying objects? $\endgroup$ Sep 25, 2019 at 20:42
  • $\begingroup$ Kashiwara crystals can be defined relative to any root system. A basic reference is the book "Crystal Bases: Representations And Combinatorics" by Bump and Schilling. The combinatorics for Type B crystals is described briefly in Section 6.3.2 of that book. $\endgroup$ Sep 25, 2019 at 21:23
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Tensor product decompositions are well studied and there are algorithms at least in all classical cases. (By the way, it is a special kind of branching rule for diagonal inclusion $G \to G\times G.$) See e.g. Wikipedia page on LR rule and references therein. https://en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson_rule

In general, it is a complicated problem with some remaining open questions. In practice, one has some particular constraints and an already chosen parametrisation (i.e. Young diagrams, highest weights, crystals, ...) and goes on from there. There is at least one case when the tensor product with fundamental representation has simple rule which goes under the name Pieri rule and that is the case of the standard representation $\mathbb{C}^n.$

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