Simple SL(n)-modules with normal closures of maximal torus orbits

Abstract

Let T be the subgroup of diagonal matrices in the group SL(n). The aim of this paper is to find all finite-dimensional simple rational SL(n)-modules V with the following property: for each point v?V the closure $\overline{Tv}$ of its T-orbit is a normal affine variety. Moreover, for any SL(n)-module without this property a T-orbit with non-normal closure is constructed. The proof is purely combinatorial: it deals with the set of weights of simple SL(n)-modules. The saturation property is checked for each subset in the set of weights.