The recording tableaux in the quantum Littlewood-Richardson map, the orthogonal transpose symmetry map, and the computation of highest weight tableaux

TL;DR

This paper explores the quantum Littlewood-Richardson map, providing a combinatorial proof of its surjectivity, and applies it to compute highest weight tableaux related to the Naito-Sagaki conjecture.

Contribution

It offers a new combinatorial proof of the surjectivity of the quantum LR map and introduces a branching model for multiplicities from GL_{2n} to Sp_{2n}.

Findings

Provides a combinatorial proof for the surjectivity of the quantum LR map.

Establishes a new branching model for multiplicities from GL_{2n} to Sp_{2n}.

Computes highest weight tableaux related to the Naito-Sagaki conjecture.

Abstract

Recently Watanabe has given an algorithm to compute a bijection, that he calls (quantum) Littlewood-Richardson (LR) map (or quantum LR rule of type AII), between semi-standard Young tableaux of shape a partition with at most $2n$ parts and pairs of tableaux consisting of a symplectic tableau with shape a partition with at most $n$ parts, and a recording tableau of skew-shape given by the two previous shapes. The recording tableaux in that algorithm are shown to be equinumerous to Littlewood-Richardson-Sundaram tableaux whose injectivity is shown combinatorially while the surjectivity is concluded via representation theory of a quantum symmetric pair of type AII$_{2n-1}$. Henceforth, the algorithm to compute the quantum LR map provides a new branching model for the branching multiplicities from $GL_{2n}(\C)$ to $Sp_{2n}(\C)$. Here, as morally suggested by Watanabe, one provides a combinatorial proof for the surjectivity of the quantum LR map which in turn exhibits the restriction of the LR orthogonal transpose symmetry map to LR-Sundaram tableaux. As an application of the explicit surjectivity, we compute certain highest weight tableaux as defined in the recent proof of the Naito-Sagaki conjecture using the Watanabe's branching rule based on the crystal basis theory for $\imath$quantumgroups of type AII$_{2n-1}$.