An observation on highest weight crystals

Abstract

As shown by Stembridge, crystal graphs can be characterized by their local behavior. In this paper, we observe that a certain local property on highest weight crystals 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. In other classical types, there is a similar (but necessarily more technical) statement. This walk is obtained from the associated level 1 perfect crystal, $B^{1,1}$. (It is unique unless the Dynkin diagram contains that of $D_4$ as a subdiagram.) This crystal observation was motivated by representation-theoretic behavior of the affine Hecke algebra of type $A$, which is known to be captured by highest weight crystals of type $A^{(1)}$ by results of Grojnowski. As discussed below, the proofs in either setting are straightforward, and so the theorem linking the two phenomena is not needed. However, the result is presented here for crystals as one can say something in all types (Grojnowski's theorem is only in type $A$), and because the statement seems more surprising in the language of crystals than it does for affine Hecke algebra modules.