Sottile, Frank2010-01-142010-01-162017-04-072010-01-142010-01-162017-04-072007-082009-05-15http://hdl.handle.net/1969.1/ETD-TAMU-1550The Drinfel?d Lagrangian Grassmannian compacti?es the space of algebraic maps of ?xed degree from the projective line into the Lagrangian Grassmannian. It has a natural projective embedding arising from the highest weight embedding of the ordinary Lagrangian Grassmannian, and one may study its de?ning ideal in this embedding.The Drinfel?d Lagrangian Grassmannian is singular. However, a concrete description of generators for the de?ning ideal of the Schubert subvarieties of the Drinfel?d Lagrangian Grassmannian would implythat the singularities are modest. I prove that the de?ning ideal of any Schubert subvariety is generated by polynomials which give a straightening law on an ordered set. Using this fact, I show that any such subvariety is Cohen-Macaulay and Koszul. These results represent a partial extension of standard monomial theory to the Drinfel?d Lagrangian Grassmannian.en-USquasimapsalgebras with straightening lawBook