Relating two crystal-theoretic descriptions of the quantum twist automorphism

Determine how the abstract crystal-theoretic description of the quantum twist automorphism in Kimura–Oya, Theorem 5.22, is connected with the explicit description in terms of PBW and string parametrizations, g-vector transformations, and the matrices M_bfi and N_bfi given in Theorem 3.8.

Background

The paper provides an explicit description of the crystal counterpart of the quantum twist automorphism using PBW and string parametrizations together with linear transformations induced by g-vectors. It also notes that Kimura and Oya previously gave an abstract crystal-theoretic description. The precise mathematical relationship between these two formulations is not clarified in the paper.