A straightening law for the Drinfel'd Lagrangian Grassmannian

Abstract

The 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.