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In recent work with Schumann we have proven a conjecture of Naito-Sagaki giving a branching rule for the decomposition of the restriction of an irreducible representation of the special linear Lie algebra to the symplectic Lie algebra, therein embedded as the fixed-point set of the involution obtained by the folding of the corresponding Dyinkin diagram. It provides a new approach to branching rules for non-Levi subalgebras in terms of Littelmann paths. In this paper we motivate this result, provide examples, and give an overview of the combinatorics involved in its proof.

On a conjecture of Naito-Sagaki: Littelmann paths and Littlewood-Richardson Sundaram tableaux