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

we consider the semigroup $s$ of highest weights appearing in tensor powers $v^{otimes k}$ 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 $v^{otimes k}$ in terms of the volume of this polytope.