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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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