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

Document Type : Research Paper

Author

Department of Mathematics, Dietrich School of Arts and Sciences, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, U.S.A.

Abstract

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.

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### History

• Receive Date: 30 October 2013
• Revise Date: 31 March 2014
• Accept Date: 06 April 2014