  The Drinfel'd Lagrangian Grassmannian compactifies the space of algebraic
maps of fixed degree from the projective line into the Lagrangian Grassmannian.
It has a natural projective embedding arising from the canonical embedding of
the Lagrangian Grassmannian. We show that the defining ideal of any Schubert
subvariety of the Drinfel'd Lagrangian Grassmannian is generated by polynomials
which give a straightening law on an ordered set. Consequentially, any such
subvariety is Cohen-Macaulay and Koszul. The Hilbert function is computed from
the straightening law, leading to a new derivation of certain intersection
numbers in the quantum cohomology ring of the Lagrangian Grassmannian.
