An observation on highest weight crystals

In this paper, we observe that a certain local property on highest weight crystal graphs forces a more global property. In type A, this statement says that if a node has a single parent and single grandparent, then there is a unique walk from the highest weight node to it. This crystal observation was motivated by certain representation-theoretic behavior of the affine Hecke algebra of type A. In other classical types, there is a similar statement. This walk is obtained from the associated level 1 perfect crystal, B1,1. (It is unique unless the Dynkin diagram contains that of D4 as a subdiagram.)