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

TL;DR

This paper proves a specific case of a conjecture relating Littelmann paths and Littlewood-Richardson Sundaram tableaux, providing insights into branching rules for certain Lie algebra representations.

Contribution

It establishes a special case of the Naito-Sagaki conjecture using Littelmann paths and suggests a broader approach via Sundaram tableaux.

Findings

Confirmed a special case of the Naito-Sagaki conjecture.

Connected Littelmann paths with Littlewood-Richardson Sundaram tableaux.

Proposed a general approach to the conjecture.

Abstract

We prove a special case of a conjecture of Naito-Sagaki about a branching rule for the restriction of irreducible representations of $\mathfrak{sl}(2n,\mathbb{C})$ to $\mathfrak{sp}(2n,\mathbb{C})$. The conjecture is in terms of certain Littelmann paths, with the embedding given by the folding of the type $A_{2n-1}$ Dynkin diagram. We propose and motivate an approach to the conjecture in general, in terms of Littlewood-Richardson Sundaram tableaux.