The inverse reduction map in the quantum Littlewood-Richardson bijection

In arXiv:2603.16698v5 we have explicitly computed the inverse of the reduction map in the quantum Littlewood-Richardson bijection for certain patterns of symplectic columns.

It turns out that some of those patterns are cell pieces to compute the inverse of the reduction map on any symplectic column.

For large symplectic columns, the tools provided here can be combined with the composition of the inverses of the several maps in which the reduction map decomposes, given by Watanabe, namely, among them, combinatorial $R$-matrices and reduction maps of shorter symplectic columns.