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 ! sln (and the loop group SLn). 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 Jimbo et. al. 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. The most important representation of the affine Kac-Moody algebra ŝln (or of the loop group ŜLn) is the basic representation V (Λ0), the highest-weight representation associated to the extra node of the extended Dynkin diagram A n−1. The infinite-dimensional space V (Λ0) is filtered by the finite-dimensional Demazure modules Vw(Λ0) for w an element of the affine Weyl group: these are modules for a Borel subgroup of the loop group. There are several general constructions for irreducible representations and their Demazure modules, such as Lusztig’s canonical basis and Littelmann’s contracting modules. However, they are extremely difficult to compute explicitly, and even the combinatorial indexing set for a basis is very intricate. We will give an elementary and easily computable basis for V (Λ0) and its Demazure modules. We work inside the Fock space F , an infinite wedge product which contains V (Λ0), analogously to the space ∧C which realizes a fundamental representation of SLnC. The Fock space has a natural basis indexed by certain infinite subsets of integers. The combinatorial part of our problem amounts to defining which of these subsets will index basis elements of Vw(Λ0) for a given w. We describe these special subsets in terms of a recursive but very simple algorithm, the roof operator on subsets. This is analogous to the left-key construction of Lascoux-Schutzenberger [11], which distinguishes the Young tableaux indexing a basis of a given Demazure module of SLnC. The roof operator is more elementary (and much more efficient) than the crystal graph operators, and is in some sense orthogonal to them. One may think of the roof operator as jumping across the crystal graph, moving each vertex down to an extremal weight vertex w(Λ0), but not along edges of the crystal graph. The combinatorics of the roof operator lead naturally to the definition of our standard basis, in analogy to the method of Raghavan-Sankaran [16]. A novel feature is that we define our basis “bottom-up” by raising each extremal weight