The inverse reduction map of a symplectic column by decreasing the rank by one

Abstract

We have previously given a factorization of a symplectic column under the action of the parity involution which enabled to explicitly have written the inverse of the reduction map in the quantum Littlewood-Richardson bijection. Watanabe has written the reduction map as a composition of several maps, among them, combinatorial $R$-matrices on single columns and a reduction map on a shorter column with rank reduced by one. We now use this approach to write the inverse of the reduction map on a symplectic column by detecting the corresponding symplectic column of rank reduced by one and thus avoiding going through several map compositions.