A crystal embedding into Lusztig data of type A

Let i be a reduced expression of the longest element in the Weyl group of type A, which is adapted to a Dynkin quiver with a single sink. We present a simple description of the crystal embedding of Young tableaux of arbitrary shape into i-Lusztig data, which also gives an algorithm for the transition matrix between Lusztig data associated to reduced expressions adapted to quivers with a single sink.