Simple SL(n)-Modules with Normal Closures of Maximal Torus Orbits

TL;DR

This paper classifies all finite-dimensional simple SL(n)-modules where the closure of each torus orbit is normal, using combinatorial methods based on weight sets, and constructs examples where this property fails.

Contribution

It provides a complete classification of simple SL(n)-modules with normal torus orbit closures and introduces a combinatorial approach to verify the saturation property of weight sets.

Findings

Identifies all simple SL(n)-modules with normal orbit closures.

Constructs examples of modules with non-normal orbit closures.

Develops a combinatorial method based on weights to analyze normality.

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 $v\in V$ the closure $\bar{Tv}$ 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.