Séminaire Lotharingien de Combinatoire, 78B.53 (2017), 11 pp.

Jacinta Torres

On a Conjecture of Naito-Sagaki: Littelmann Paths and Littlewood-Richardson Sundaram Tableaux

Abstract. 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 Dynkin diagram. This conjecture had been open for over ten years, and provides a new approach to branching rules for non-Levi subalgebras in terms of Littelmann paths. In this extended abstract we motivate the conjecture, prove it for several cases, where we also relate it to the combinatorics of polytopes and Littlewood-Richardson cones, and highlight some difficulties of the proof in general.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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