Bi-crystals and crystal (GL(V ), GL(W )) duality

Consider the tensor product of finite dimensional vector spaces V ⊗ W . We have an action of GL(V ) on V ⊗W induced by standard action on V . Similarly the action of GL(W ) on W gives us an action on V ⊗W . These actions of GL(V ) and GL(W ) on V ⊗W clearly commute with one another, so we have a joint action of GL(V )×GL(W ) on V ⊗W . Let S•(V ) and Λ•(V ) denote the symmetric and exterior algebras on V , respectively. It is standard that the action of GL(V ) gives rise to an action on S•(V ) and Λ•(V ) by algebra homomorphism. We can consider the restriction of the action of GL(V ⊗W ) on S•(V ⊗W ) (or Λ•(V ⊗W )) to GL(V ) × GL(W ). For this action we have explicit decompositions of S•(V ⊗W ) and Λ•(V ⊗W ) into irreducible modules (see, for example [5]): S•(V ⊗W ) ∼= ∑