Standard Bases for Affine SL(n)-Modules

Abstract

We give an elementary and easily computable basis for the Demazure modules in the basic representation of the affine Lie algebra sl(n)-hat (and the loop group SL(n)-hat). A novel feature is that we define our basis ``bottom-up'' by raising each extremal weight vector, rather than ``top-down'' by lowering the highest weight vector. Our basis arises naturally from the combinatorics of its indexing set, which consists of certain subsets of the integers first specified by the Kyoto school in terms of crystal operators. We give a new way of defining these special sets in terms of a recursive but very simple algorithm, the roof operator, which is analogous to the left-key construction of Lascoux-Schutzenberger. The roof operator is in a sense orthogonal to the crystal operators.