Tensoring irreducible $B$-series representations/ Type B Littlewood-Richardson 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.
 A: 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.
A: 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.
A: 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 $\mathbb{C}^n.$
