The recording tableaux in the quantum Littlewood-Richardson map, the orthogonal transpose symmetry map, and the computation of $\mathfrak{k}$-highest weight tableaux

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. The surjectivity is exhibited via the reverse Schensted insertion on the quantum recording tableaux, ruled by the slack data, followed with the inverse of the reduction map on the bumped entries that we explicitly compute for certain families of symplectic columns. As an application of the inverse of the quantum LR map, we compute and characterize by certain linear inequalities a family of $\mathfrak{k}$- highest weight semi-standard tableaux in the recent proof of the Naito-Sagaki conjecture using the Watanabe'sbranching rule based on the crystal basis theory for $\imath$quantumgroups of type AII$_{2n-1}$.