A remark on asymptotic enumeration of highest weights in tensor powers of a representation

TL;DR

This paper analyzes the asymptotic behavior of highest weights in tensor powers of a representation, linking their distribution to the volume of the associated weight polytope.

Contribution

It provides a geometric description of the cone of highest weights and derives asymptotic counts based on the volume of the weight polytope.

Findings

The cone of highest weights is the cone over the weight polytope intersected with the positive Weyl chamber.

Asymptotic number of highest weights is proportional to the volume of the weight polytope.

The results connect representation theory with geometric and combinatorial structures.

Abstract

We consider the semigroup S of highest weights appearing in tensor powers V^k of a finite dimensional representation V of a connected reductive group. We describe the cone generated by S as the cone over the weight polytope of V intersected with the positive Weyl chamber. From this we get a description for the asymptotic of the number of highest weights appearing in V^k in terms of the volume of this polytope.