Symplectic Analogs of the Distributive Lattices L(m, n)

We introduce two families of symplectic analogs of the distributive lattices L(m, n). We give several combinatorial descriptions of these distributive lattices and use combinatorial methods to produce their rank generating functions. Using Proctorʹs sl(2, C) technique, we prove that these symplectic lattices are rank symmetric, rank unimodal, and strongly Sperner. This confirms a conjecture of Reiner and Stanton concerning one of these families of symplectic lattices. We describe how both families of symplectic lattices can be used to explicitly realize the fundamental representations of the symplectic Lie algebras.