  Truncated shifted Yangians are a family of algebras which are natural
quantizations of slices in the affine Grassmannian. We study the highest weight
representations of these algebras. In particular, we conjecture that the
possible highest weights for these algebras are described by product monomial
crystals, certain natural subcrystals of Nakajima's monomials. We prove this
conjecture in type A. We also place our results in the context of symplectic
duality and prove a conjecture of Hikita in this situation.
